1.1. Definitions

1.1.1. Model and empirical measure

We consider the probability measure $\mu _{N,\beta }^{V;\mathsf {B}}$ on $\mathsf {B}^N$ given by

(1.1) $$ \begin{align} \mathrm{d}\mu_{N,\beta}^{V;\mathsf{B}}(\lambda) = \frac{1}{Z_{N,\beta}^{V;\mathsf{B}}}\prod_{i = 1}^N \mathrm{d}\lambda_i\,\mathbf{1}_{\mathsf{B}}(\lambda_i)\,e^{-\frac{\beta N}{2} \,V(\lambda_i)}\,\prod_{1 \leq i < j \leq N} |\lambda_i - \lambda_j|^{\beta}. \end{align} $$

$\mathsf {B}$ is a finite disjoint union of closed intervals of $\mathbb {R}$ possibly with infinite endpoints, $\beta $ is a positive number and $Z_{N,\beta }^{V;\mathsf {B}}$ is the partition function so that (1.1) has total mass $1$ . This model is usually called the $\beta $ -ensemble [Reference MehtaMeh04, Reference Dumitriu and EdelmanDE02, Reference ForresterFor10]. We introduce the unnormalised empirical measure $M_N$ of the eigenvalues

$$ \begin{align*}M_N=\sum_{i=1}^N \delta_{\lambda_i}, \end{align*} $$

and we consider several types of statistics for $M_N$ . We sometimes denote $\mathbb {L} = \mathrm {diag}(\lambda _1,\ldots ,\lambda _N)$ .

1.1.2. Correlators

We introduce the Stieltjes transform of the n-th order moments of the empirical measure, called disconnected correlators:

$$ \begin{align*}\widetilde{W}_n(x_1,\ldots,x_n) = \mu_{N,\beta}^{V;\mathsf{B}}\Big[\Big(\int_{\mathbb{R}}\frac{\mathrm{d} M_N(\xi_1)}{x_1-\xi_1}\cdots\int_{\mathbb{R}}\frac{\mathrm{d} M_N(\xi_n)}{x_n - \xi_n}\Big)\Big]. \end{align*} $$

They are holomorphic functions of $x_i \in \mathbb {C}\setminus \mathsf {B}$ . It is more convenient to consider the correlators to study large N asymptotics:

(1.2) $$ \begin{align} \nonumber W_n(x_1,\ldots, x_n) & = \partial_{t_1}\cdots\partial_{t_n}\Big(\ln Z_{N,\beta}^{V-\frac{2}{\beta N}\sum_{i = 1}^n \frac{t_i}{x_i - \bullet};\mathsf{B}}\Big)\Big|_{t_i = 0} \\ & = \mu_{N,\beta}^{V;\mathsf{B}}\Big[\prod_{i = 1}^n {\,\mathrm{Tr}}\:\,\frac{1}{x_j - \mathbb{L}}\Big]_{c}. \end{align} $$

By construction, the coefficients of their expansions as a Laurent series in the variables $x_i $ (sufficiently large) give the n-th order cumulants of $M_N$ . If I is a set, we introduce the notation $x_I = (x_i)_{i \in I}$ for a set of variables indexed by I; their order will not matter as we insert them only in symmetric functions of their variables (like $W_n$ , $\widetilde {W}_n$ , etc.). The two types of correlators are related by

where $\dot {\cup }$ stands for the disjoint union. If $\varphi _n$ is an analytic (symmetric) function in n variables in a neighbourhood of $\mathsf {B}^n$ , then the n-linear statistics can be deduced as contour integrals of the disconnected correlators:

(1.3) $$ \begin{align} \mu_{N,\beta}^{V;\mathsf{B}}\Big[ \sum_{1 \leq i_1,\ldots,i_n \leq N} \varphi_n(\lambda_{i_1},\ldots,\lambda_{i_n})\Big] = \oint_{\mathsf{B}} \frac{\mathrm{d} \xi_1}{2\mathrm{i}\pi} \cdots \oint_{\mathsf{B}} \frac{\mathrm{d} \xi_n}{2\mathrm{i}\pi}\,\varphi_n(\xi_1,\ldots,\xi_n)\,\widetilde{W}_n(\xi_1,\ldots,\xi_n). \end{align} $$

We remark that the knowledge of the correlators for an analytic family of potentials $(V_{t})_{t}$ determines the partition function up to an integration constant since

$$ \begin{align*}\partial_t \ln Z_{N,\beta}^{V_t;\mathsf{B}} = -\frac{\beta N}{2}\,\mu_{N,\beta}^{V_t;\mathsf{B}}\Big[\sum_{i = 1}^N \partial_t V_t(\lambda_i)\Big] = -\frac{\beta N}{2}\,\oint_{\mathsf{B}} \frac{\mathrm{d}\xi}{2\mathrm{i}\pi}\,\partial_t V_t(\xi)\,W_1^t(\xi), \end{align*} $$

where $W_1^{t}$ is the first correlator in the model with potential $V_t$ , and the notation $\oint _{\mathsf {B}} \mathrm {d} \xi \cdots $ means integration along a contour in $\mathbb {C} \setminus \mathsf {B}$ surrounding $\mathsf {B}$ with positive orientation. If the integrand has poles in $\mathbb {C} \setminus \mathsf {B}$ (e.g., it depends on extra variables $x_i \in \mathbb {C} \setminus \mathsf {B}$ that are not integrated upon and has poles at $\xi = x_i$ ), the contour should be chosen (unless stated otherwise) so that the poles remain outside. The notation should not be confused with $\int _{\mathsf {B}} \mathrm {d} \xi \cdots $ , which is the Lebesgue integral on $\mathsf {B} \subseteq \mathbb {R}$ .

1.1.3. Kernels

Let $\mathbf {c}$ be a n-tuple of nonzero complex numbers. We introduce the n-point kernels:

(1.4) $$ \begin{align} \nonumber \mathsf{K}_{n,\mathbf{c}}(x_1,\ldots,x_n) & = \mu_{N,\beta}^{V;\mathsf{B}}\left[\prod_{j = 1}^n \mathrm{det}^{c_j}(x_j - \mathbb{L})\right] \\ & = \frac{Z_{N,\beta}^{V - \frac{2}{\beta N}\sum_{j = 1}^n c_j\ln(x_j - \bullet);\mathsf{B}}}{Z_{N,\beta}^{V;\mathsf{B}}}. \end{align} $$

When $c_j$ are integers, the kernels are holomorphic functions of $x_j \in \mathbb {C}\setminus \mathsf {B}$ . When $c_j$ are not integers, the kernels are multivalued holomorphic functions of $x_j$ in $\mathbb {C}\setminus \mathsf {B}$ , with monodromies around the connected components of $\mathsf {B}$ and around $\infty $ . The right-hand side of (1.4), where we used $\ln $ , has the same multivalued nature. Alternatively, both sides of (1.4) can be defined as single-valued functions of $x_1,\ldots ,x_n$ by choosing a determination of the logarithm in a domain $\mathsf {D}$ of the form $\mathbb {C} \setminus \ell $ , where $\ell $ is a smooth path in $\mathbb {C}$ from $0$ to $\infty $ , and using $z^{c} = e^{c \ln z}$ for the left-hand side.

In particular, for $\beta = 2$ , $\mathsf {K}_{1,(1)}(x)$ is the monic N-th orthogonal polynomial associated to the weight $\mathbf {1}_{\mathsf {B}}(x)\,e^{-N\,V(x)}\mathrm {d} x$ on the real line, and $\mathsf {K}_{2,(1,-1)}(x,y)$ is the N-th Christoffel–Darboux kernel associated to those orthogonal polynomials; see Section 2.

1.2. Equilibrium measure and multi-cut regime

By standard results of potential theory and large deviations – see [Reference JohanssonJoh98, Reference Ben Arous and GuionnetBAG97] or the textbooks [Reference DeiftDei99, Theorem 6] or [Reference Anderson, Guionnet and ZeitouniAGZ10, Theorem 2.6.1 and Corollary 2.6.3] (note there that $\mathsf {B}=\mathbb R$ , but the generalisation to integration over general sets $\mathsf {B}$ is straightforward) – we have the following:

Theorem 1.1. Assume that $V\,:\, \mathsf {B} \rightarrow \mathbb {R}$ is a continuous function, and if V depends on N, assume also that $V $ converges towards $V^{\{0\}}$ when N goes to infinity in the space of continuous functions over $\mathsf {B}$ for the sup norm. Moreover, for $\tau \in \{\pm 1\}$ with $\tau \infty \in \mathsf {B}$ , assume that

$$ \begin{align*}\liminf_{x \rightarrow \tau\infty} \frac{V^{\{0\}}(x)}{2\ln|x|}> 1. \end{align*} $$

We consider the normalised empirical measure $L_N=N^{-1}\,M_N$ in the space $\mathcal {P}(\mathsf {B})$ of probability measures on $\mathsf {B}$ equipped with its weak topology. Then, the law of $L_N$ under $\mu _{N,\beta }^{V;\mathsf {B}}$ satisfies a large deviation principle with scale $N^2$ and good rate function J given by

(1.5) $$ \begin{align} J[\mu]= E[\mu]-\inf_{\nu\in\mathcal{P}(\mathsf{B})} E[\nu],\qquad E[\mu] =\frac{\beta}{2} \iint_{\mathsf{B}^2} \mathrm{d}\mu(\xi)\mathrm{d}\mu(\eta)\Big(\frac{V^{\{0\}}(\xi) + V^{\{0\}}(\eta)}{2} -\ln|\xi-\eta|\Big). \end{align} $$

As a consequence, $L_N$ converges almost surely and in expectation to the unique probability measure $\mu _\mathrm{{eq}}^{V}$ on $\mathsf {B}$ which minimises E. $\mu _\mathrm{{eq}}^{V}$ has compact support, denoted $\mathsf {S}$ . It is characterised by the existence of a constant $C^V$ such that

(1.6) $$ \begin{align} \forall x \in \mathsf{B},\qquad 2\int_{\mathsf{B}}\mathrm{d} \mu_{\mathrm{eq}}^{V}(\xi)\ln|x - \xi| - V^{\{0\}}(x) \leq C^V, \end{align} $$

with equality realised $\mu _\mathrm{{eq}}^V$ almost surely.

The goal of this article is to establish an all-order expansion of the partition function, the correlators and the kernels in all such situations.

1.3. Assumptions

We will refer throughout the text to the following set of assumptions. An integer number $g\ge 0$ is fixed.

Note that if $V^{\{0\}}$ is real-analytic in a neighbourhood of $\mathsf {B}$ , the $(g + 1)$ -cut regime hypothesis is always satisfied (the support consists of a finite disjoint union of segments) and S is analytic in a neighbourhood of $\mathsf {S}$ . We will hereafter say that V is regular and confining in $\mathsf {B}$ if it satisfies the two first assumptions above. We will also require a stronger regularity for the potential.

In Section 6, we shall weaken Hypothesis 1.2 by allowing complex perturbations of order $\frac {1}{N}$ and harmonic functions instead of analytic functions.

The topology for which we study the large N expansion of correlators is described in § 5 and amounts to controlling the (moments of order p) $\times C^p$ uniformly in p for a constant $C> 0$ . We now describe our strategy and announce our results.

1.4. Main result with fixed filling fractions: partition function and correlators

Before coming to the multi-cut regime, we analyse a different model where the number of $\lambda $ s in a small enlargement of $\mathsf {S}_h$ is fixed. Let $\mathsf {A} = \bigcup _{h = 0}^{g} \mathsf {A}_h$ , where $\mathsf {A}_h = [a_{h}^{-},a_h^{+}]$ are pairwise disjoint segments such that $a_{h}^{-} \leq \alpha _{h}^{-} < \alpha _{h}^+ \leq a_{h}^+$ , where the inequalities are equalities if the corresponding edge is hard and are strict if the corresponding edge is soft. We introduce the set

(1.9) $$ \begin{align} \mathcal{E} = \Big\{\boldsymbol{\epsilon} \in (0,1)^{g}\quad\Big| \quad \sum_{h = 1}^{g} \epsilon_h < 1\Big\}. \end{align} $$

If $\boldsymbol {N}=(N_1,\ldots ,N_g)$ is an integer vector such that $\boldsymbol {\epsilon }=\frac {\boldsymbol {N}}{N} \in \mathcal {E}$ , we denote $N_0 = N - \sum _{h = 1}^{g} N_h$ and consider the probability measure on $\prod _{h = 0}^{g} \mathsf {A}_h^{N_h}$ :

(1.10) $$ \begin{align} \nonumber \mathrm{d}\mu_{N,\beta;\boldsymbol{\epsilon}}^{V;\mathsf{A}}(\boldsymbol{\lambda}) & = \frac{1}{Z_{N,\beta;\boldsymbol{\epsilon}}^{V;\mathsf{A}}}\prod_{h = 0}^g \Big[\prod_{i = 1}^{N_h} \mathrm{d}\lambda_{h,i}\,\mathbf{1}_{\mathsf{A}_{h}}(\lambda_{h,i})\,e^{-\frac{\beta N}{2}\,V(\lambda_{h,i})}\,\prod_{1 \leq i < j \leq N} |\lambda_{h,i} - \lambda_{h,j}|^{\beta}\Big] \\ & \quad \times \prod_{0 \leq h < h' \leq g} \prod_{\substack{1 \leq i \leq N_h \\ 1 \leq i' \leq N_{h'}}} |\lambda_{h,i} - \lambda_{h',i'}|^{\beta}. \end{align} $$

The empirical measure $M_{N}$ and the correlators $W_{n;\boldsymbol {N}/N}(x_1,\ldots ,x_n)$ for this model are defined as in § 1.1 with $\mu _{N,\beta }^{V;\mathsf {A}}$ replaced by $\mu _{N,\beta ;\boldsymbol {N}/N}^{V;\mathsf {A}}$ . We call $\epsilon _h=\frac {N_h}{N}$ the filling fraction of $\mathsf {A}_h$ . It follows from the definitions that

(1.11) $$ \begin{align} \oint_{\mathsf{A}_h} \frac{\mathrm{d}\xi}{2\mathrm{i}\pi}\,W_{n;\boldsymbol{N}/N}(\xi,x_2,\ldots,x_n) = \delta_{n,1}\,N_{h}=\delta_{n,1}\,N \epsilon_{h} \end{align} $$

for $x_2,\ldots ,x_n \in \mathbb {C} \setminus \mathsf {A}$ . Indeed, from the definition of the correlators (1.2), $W_{n;\boldsymbol {N}/N}(x_{1},x_2,\ldots ,x_n) $ for $n \geq 2$ can be expressed as a sum of products of moments of products of the n-tuple of random variables $\big (\sum _{i = 1}^{N} \frac {1}{x_j - \lambda _{i}}-\mu _{N,\beta ;\boldsymbol {\epsilon }}^{V;\mathsf {A}}[\sum _{i = 1}^{N} \frac {1}{x_j - \lambda _{i}}] \big )_{j = 1}^{n}$ which are linear in each of these variables. Therefore, we can integrate over the variable $x_{1}$ in each of these terms by Fubini’s theorem. The key observation is that $\oint _{\mathsf {A}_h} \sum _{i = 1}^{N} \frac {\mathrm{d} \xi }{2\mathrm{i}\pi }\,\frac {1}{\xi - \lambda _i}$ is the number $N_{h}$ of $\lambda _i$ s belonging to $\mathsf {A}_h$ . Since $N_h$ is deterministic in the fixed filling fraction model, it is equal to its expectation, and therefore, each of these terms vanish which implies (1.11) for $n\ge 2$ . When $n=1$ , the cumulant is simply equal to the expectation of $ \sum _{i = 1}^{N} \frac {1}{\xi - \lambda _i}$ , and the previous remark proves (1.11).

We will refer to (1.1) as the initial model and to (1.10) as the model with fixed filling fractions. Standard results from potential theory or a straightforward generalisation of [Reference Anderson, Guionnet and ZeitouniAGZ10, Theorem 2.6.1 and Corollary 2.6.3] imply the following:

Theorem 1.2. Assume V regular and confining on $\mathsf {A}$ . We consider the normalised empirical measures $L_{N,h}=\frac {1}{N_h}\sum _{i=1}^{N_h} \delta _{\lambda _{h,i}}\in \mathcal {P}(\mathsf {A}_h)$ for . Take a sequence $\boldsymbol {N} = (N_1,\ldots ,N_g)$ of g-tuple of integers, indexed by N, such that $\sum _{h = 1}^g N_h \leq N$ , and such that $\boldsymbol {N}/N$ converges to a given $\boldsymbol {\epsilon }\in \mathcal {E}$ when $N \rightarrow \infty $ . Then, the law of $(L_{N,h})_{0\le h\le g}$ under $\mu _{N,\beta ;\boldsymbol {N}/N}^{V;\mathsf {A}}$ satisfies a large deviation principle with scale $N^2$ and good rate function

$$ \begin{align*}J_{\boldsymbol{\epsilon}}[\mu_0,\ldots,\mu_g]=E\Big[\sum_{h=0}^g\epsilon_h \mu_h\Big]-\inf_{\nu_h\in\mathcal{P} (\mathsf{A}_h)} E\Big[\sum_{h=0}^g\epsilon_h \nu_h\Big]\,,\end{align*} $$

where $\epsilon _0=1-\sum _{h=1}^g \epsilon _h$ , $N_0=N-\sum _{h=1}^g N_h$ and E is defined in Equation (1.5). As a consequence, the empirical measure $L_{N;\boldsymbol {\epsilon }}=\sum _{h=0}^g\frac {N_h}{N} L_{N,h}$ converges almost surely and in expectation towards the unique probability measure $\mu _\mathrm{{eq};\boldsymbol {\epsilon }}^{V;\mathsf {A}}$ on $\mathsf {A}$ which minimises E among probability measures with fixed mass $\epsilon _h$ on $\mathsf {A}_h$ for any . It is characterised by the existence of constants $C_{\boldsymbol {\epsilon },h}^{V,\mathsf {A}}$ such that

(1.12)

with equality realised $\mu _\mathrm{{eq};\boldsymbol {\epsilon }}^{V;\mathsf {A}}$ almost surely. $\mu _\mathrm{{eq};\boldsymbol {\epsilon }}^{V;\mathsf {A}}$ can be decomposed as a sum of positive measures $\mu _\mathrm{{eq};\boldsymbol {\epsilon },h}^{V}$ having compact support in $\mathsf {A}_h$ , denoted $\mathsf {S}_{\boldsymbol {\epsilon },h}$ . Moreover, if $V^{\{0\}}$ is real-analytic in a neighbourhood of $\mathsf {A}$ , the support $\mathsf {S}_{\boldsymbol {\epsilon },h}$ consists of a finite union of segments.

Later in the text, we shall consider $\mu _\mathrm{{eq};\boldsymbol {N}/N}^{V;\mathsf {A}}$ with $\boldsymbol {N}=(N_1,\ldots ,N_g)$ a vector of positive integers so that $\sum _{h = 1}^{g} N_h < N$ : this will denote the unique solution of (1.12) with $\boldsymbol {\epsilon }=\boldsymbol {N}/N$ . $\mu _\mathrm{{eq}}^{V;\mathsf {A}}$ appearing in Theorem 1.1 coincides with $\mu _\mathrm{{eq};\boldsymbol {\epsilon }_{\star }}^{V}$ for the optimal value $\boldsymbol {\epsilon }_{\star } = (\mu _\mathrm{{eq}}^{V;\mathsf {A}}(\mathsf {A}_{h}))_{1 \leq h \leq g}$ , and in this case, $\mathsf {S}_{\boldsymbol {\epsilon }_{\star },h}$ is actually the segment $[\alpha ^{-}_h,\alpha _{h}^+]$ . The key point – justified in Appendix 1 – is that, for $\boldsymbol {\epsilon }$ close enough to $\boldsymbol {\epsilon }_{\star }$ , the support $\mathsf {S}_{\boldsymbol {\epsilon },h}$ remains connected, and the model with fixed filling fractions enjoys a $\frac {1}{N}$ expansion.

Theorem 1.3. If V satisfies Hypotheses 1.1 and 1.3 on $\mathsf {A}$ , there exists $t> 0$ such that, uniformly for integers $\boldsymbol {N}=(N_1,\ldots ,N_g)$ such that $\boldsymbol {N} /N\in \mathcal {E}$ and $|\boldsymbol {N}/N- \boldsymbol {\epsilon }_{\star }|_1 < t$ , we have an expansion for the correlators, for any $K \geq 0$ ,

(1.13) $$ \begin{align} W_{n;\boldsymbol{N}/N}(x_1,\ldots,x_n) = \sum_{k = n - 2}^{K} N^{-k}\,W_{n;\boldsymbol{N}/N}^{\{k\}}(x_1,\ldots,x_n) + O(N^{-(K + 1)}). \end{align} $$

Up to a fixed $O(N^{-(K + 1)})$ and for a fixed n, Equation (1.13) holds uniformly for $x_1,\ldots ,x_n$ in compact regions of $\mathbb {C}\setminus \mathsf {A}$ . The $W_{n;\boldsymbol {\epsilon }}^{\{k\}}$ can be extended into smooth functions of $\boldsymbol {\epsilon }\in \mathcal {E}$ close enough to $\boldsymbol {\epsilon }_{\star }$ .

We prove this theorem, independently of the nature soft/hard of the edges, in Section 5 for real-analytic potential (i.e., Hypothesis 1.2 instead of 1.3). For $\beta = 2$ and potential V independent of N, the coefficients of expansion $W_{n;\boldsymbol {N}/N}^{\{k\}} = 0$ are zero for $k = (n + 1) \,\,\mathrm{mod} 2$ , as is well known for hermitian random matrix models (see (1.16) and the remarks on $\beta $ -dependence in Section 1.5). The result is extended to harmonic potentials (i.e., Hypothesis 1.3) in Section 6.1. In Proposition 5.6, we provide an explicit control of the errors in terms of the distance of $x_1,\ldots ,x_k$ to $\mathsf {A}$ , and its proof makes clear that the expansion of the correlators is not expected to be uniform for $x_1,\ldots ,x_n$ chosen in a compact of $\mathbb {C}\setminus \mathsf {A}$ independently of n and K (namely, it is uniform only for K fixed). Note that we will sometimes omit to specify the dependence in $\mathsf {A},V$ , etc. in the notations (e.g., for the equilibrium measure, for the correlators and their coefficient of expansions), but we will at least include it when this dependence is of particular importance.

We then compute in Section 7 the expansion of the partition function, thanks to the expansion of $W_{1;\boldsymbol {N}/N}$ and $W_{2;\boldsymbol {N}/N}$ , by an interpolation that reduces the strength of pairwise interactions between eigenvalues in different segments while preserving the equilibrium measure. At the end of the interpolation, we are left with a product of $(g + 1)$ partition functions in a one-cut regime, for which the asymptotic expansion was established in [Reference Borot and GuionnetBG11].

Theorem 1.4. If V satisfies Hypotheses 1.1 and 1.3 on $\mathsf {A}$ , there exists $t> 0$ such that, uniformly for g-dimensional vectors of positive integers $\boldsymbol {N}$ such that $\boldsymbol {N} /N\in \mathcal {E}$ and $|\boldsymbol {N}/N- \boldsymbol {\epsilon }_{\star }|_1 < t$ , we have for any $K \geq 0$ ,

(1.14) $$ \begin{align} \frac{N!\,Z_{N,\beta;\boldsymbol{N}/N}^{V;\mathsf{A}}}{\prod_{h = 0}^{g} N_h!}= N^{\frac{\beta}{2}N + \varkappa}\exp\Big(\sum_{k = -2}^{K} N^{-k}\,F^{\{k\};V}_{\beta;\boldsymbol{N}/N} + O(N^{-(K + 1)})\Big), \end{align} $$

with

$$ \begin{align*}\varkappa = \frac{1}{2} + (\# \mathrm{soft} + 3\#\mathrm{hard})\frac{-3 + \beta/2 + 2/\beta}{24}. \end{align*} $$

Besides, $F^{\{k\};V}_{\beta ;\boldsymbol {\epsilon }}$ extends to a smooth function of $\boldsymbol {\epsilon }$ close enough to $\boldsymbol {\epsilon }_{\star }$ , and at the value $\boldsymbol {\epsilon } = \boldsymbol {\epsilon }_{\star }$ , the first derivatives of $F^{\{-2\};V}_{\beta ;\boldsymbol {\epsilon }}$ vanish and its Hessian is negative definite.

We can identify explicitly the following:

(1.15) $$ \begin{align} \nonumber F^{\{-2\};V}_{\beta;\boldsymbol{\epsilon}} & = \frac{\beta}{2}\bigg(\iint_{\mathsf{A}^2} \ln|x - y|\,\mathrm{d}\mu_{\mathrm{eq};\boldsymbol{\epsilon}}^{V}(x)\mathrm{d}\mu_{\mathrm{eq};\boldsymbol{\epsilon}}^{V}(y) - \int_{\mathsf{A}} V^{\{0\}}(x)\mathrm{d}\mu_{\mathrm{eq};\boldsymbol{\epsilon}}^{V}(x)\bigg) = -\frac{\beta}{2}\,\inf_{\nu_h \in \mathcal{P}(\mathsf{A}_h)} E\Big[\sum_{h = 0}^{g} \epsilon_h\nu_h\Big], \\ F^{\{-1\};V}_{\beta;\boldsymbol{\epsilon}} & = - \frac{\beta}{2} \int_{\mathsf{A}} V^{\{1\}}(x)\mathrm{d}\mu_{\mathrm{eq};\boldsymbol{\epsilon}}^{V}(x) + \Big(1 - \frac{\beta}{2}\Big)\Big(\mathrm{Ent}[\mu_{\mathrm{eq};\boldsymbol{\epsilon}}^{V}] - \ln\big(\tfrac{\beta}{2}\big)\Big) +\frac{\beta}{2}\ln\big(\tfrac{2\pi}{e}\big) - \ln\Gamma\big(\tfrac{\beta}{2}\big), \end{align} $$

where

$$ \begin{align*}\mathrm{Ent}[\mu] = -\int_{\mathbb{R}} \ln\Big(\frac{\mathrm{d}\mu}{\mathrm{d} x}\Big) \mathrm{d} \mu(x) \end{align*} $$

is the entropy. The formula for $F^{\{-2\};V}_{\beta ;\boldsymbol {\epsilon }}$ is obvious from potential theory, while the formula for $F^{\{-1\};V}_{\beta ;\boldsymbol {\epsilon }}$ is established in Proposition 7.1 (the first term comes from the fact that we let the potential depend on N). The appearance of the entropy in the term of order N in the free energy is well known in the one-cut case, and here we prove that it appears in the same way for the multi-cut case with fixed filling fractions, and we determine the additional constant. The term $\frac {\beta }{2} N \ln N$ is universal, while the term $\varkappa \ln N$ only depends only on the nature of the endpoints of the support. These logarithmic corrections can already be observed in the asymptotic expansion of Selberg integrals for large N computing the partition function of the classical Jacobi, Laguerre or Gaussian $\beta $ -ensembles, corresponding to a one-cut regime [Reference Borot and GuionnetBG11]. The fact that the coefficient of $\ln N$ shadows in some way the geometry of the support was observed in other contexts (see, for example, [Reference Cardy and PeschelCP88]) and is not specific to two-dimensional Coulomb gases living on a line. Their identification in the multi-cut regime and fixed filling fractions results from an interpolation with a product of one such model for each cut, which changes only the coefficients of powers of N. Up to a given $O(N^{-K})$ , all expansions are uniform with respect to the parameters of the potential and of $\boldsymbol {\epsilon }$ chosen in a compact set so that the assumptions hold. Theorems 1.3–1.4 are the generalisations to the fixed filling fractions model of our earlier results about existence of the $\frac {1}{N}$ expansion in the one-cut regime [Reference Borot and GuionnetBG11] (see also [Reference JohanssonJoh98, Reference Albeverio, Pastur and ShcherbinaAPS01, Reference Ercolani and McLaughlinEM03, Reference Bleher and ItsBI05, Reference Guionnet and Maurel-SegalaGMS07, Reference Kriecherbauer and ShcherbinaKS10] for earlier results concerning the one-cut regime in $\beta = 2$ or general $\beta $ -ensembles).

1.5. Relation with Chekhov–Eynard–Orantin topological recursion

Once these asymptotic expansions are shown to exist, by consistency, their coefficients $W_{n;\boldsymbol {\epsilon }}^{\{k\}}$ are computed by the $\beta $ topological recursion of Chekhov and Eynard [Reference Chekhov and EynardCE06]. As a matter of fact, the asymptotic expansion

$$ \begin{align*}W_{n;\boldsymbol{\epsilon}}(x_1,\ldots,x_n) = \sum_{k \geq -1} N^{-k}\,W_{n;\boldsymbol{\epsilon}}^{\{k\}}(x_1,\ldots,x_n) \end{align*} $$

has a finer structure so that for $n \geq 1$ and $k \geq -1$ , we can write

(1.16) $$ \begin{align} W_{n;\boldsymbol{\epsilon}}^{\{k\}}(x_1,\ldots,x_n) = \sum_{G = 0}^{\lfloor \frac{k - n}{2} \rfloor + 1} \Big(\frac{\beta}{2}\Big)^{1 - n - G}\Big(1 - \frac{2}{\beta}\Big)^{k + 2 - 2G - n}\,\mathcal{W}_{n;\boldsymbol{\epsilon}}^{[G,k + 2 - 2G - n]}(x_1,\ldots,x_n), \end{align} $$

where $\mathcal {W}_{n;\boldsymbol {\epsilon }}^{[G,l]}$ are the quantities computed by the topological recursion of [Reference Chekhov and EynardCE06]. The initial data consists of the nondecaying terms in the correlators – namely,

$$ \begin{align*} W_{1;\boldsymbol{\epsilon}}^{\{-1\}}(x) & = \mathcal{W}_{1;\boldsymbol{\epsilon}}^{[0,0]}(x) ,\\ W_{1;\boldsymbol{\epsilon}}^{\{0\}}(x) & = \Big(1 - \frac{2}{\beta}\Big)\mathcal{W}_{1;\boldsymbol{\epsilon}}^{[0,1]}(x), \\ W_{2;\boldsymbol{\epsilon}}^{\{0\}}(x_1,x_2) & = \frac{2}{\beta}\,\mathcal{W}_{2;\boldsymbol{\epsilon}}^{[0,0]}(x_1,x_2). \end{align*} $$

All these quantities have an analytic continuation in the variables $x_i$ on the same Riemann surface $\mathcal {C}_{\boldsymbol {\epsilon }}$ called spectral curve. The curve $\mathcal {C}_{\boldsymbol {\epsilon }}$ can, in fact, be defined as the maximal Riemann surface on which $W_{1;\boldsymbol {\epsilon }}^{\{-1\}}(x)$ , initially defined for $x \in \mathbb {C} \setminus \mathsf {A}$ , admits an analytic continuation (cf. Section 1.7 for a continued discussion on geometry of spectral curves). The information carried by the decomposition (1.16) is that, if V is chosen independent of $\beta $ and N, all the $\mathcal {W}_{n;\boldsymbol {\epsilon }}^{[G,K]}$ are also independent of $\beta $ and N (except perhaps through the implicit dependence in N of $\boldsymbol {\epsilon }$ ), and thus, the coefficients of the expansions of the correlators display a remarkable structure of Laurent polynomial in $\frac {\beta }{2}$ . This property comes from the structure of the Dyson–Schwinger equations.

From the same initial data, Chekhov and Eynard also define numbers $W_{0;\boldsymbol {\epsilon }}^{[G,K]} = \mathcal {F}_{\boldsymbol {\epsilon }}^{[G,K]}$ , which give the coefficients of the asymptotic expansion of the free energy $\ln Z_{N,\beta ;\boldsymbol {N}/N}^{V;\mathsf {A}}$ up to an integration constant independent of the potential, and which are independent of $\beta $ provided V is chosen independent of $\beta $ . More precisely, we mean that for any two potentials V and $\tilde {V}$ satisfying the assumptions of Theorem 1.4 and leading to a $(g + 1)$ -cut regime, we must have for $k \geq -2$ , by consistency with [Reference Chekhov and EynardCE06],

$$ \begin{align*}F^{\{k\};V}_{\beta;\boldsymbol{\epsilon}} - F^{\{k\};\tilde{V}}_{\beta;\boldsymbol{\epsilon}} = \sum_{G = 0}^{\lfloor \frac{k}{2} \rfloor + 1} \Big(\frac{\beta}{2}\Big)^{1 - G}\Big(1 - \frac{2}{\beta}\Big)^{k + 2 - 2G}\,\big(\mathcal{F}^{[G,k + 2 - 2G];V}_{\boldsymbol{\epsilon}} - \mathcal{F}^{[G,k + 2 - 2G];\tilde{V}}_{\boldsymbol{\epsilon}}\big). \end{align*} $$

In particular, the topological recursion defines $\mathcal {F}^{[0,0];V}_{\boldsymbol {\epsilon }} = E[\mu _{\mathrm{eq};\boldsymbol {\epsilon }}^V]$ and $\mathcal {F}^{[0,1];V}_{\boldsymbol {\epsilon }} = -\mathrm{Ent}[\mu _{\mathrm{eq};\boldsymbol {\epsilon }}^{V}]$ . By comparison with (1.15), we arrive to an absolute comparison (here, assume the potential to be independent of N – i.e., $V = V^{\{0\}}$ ):

(1.17) $$ \begin{align} \nonumber F^{\{-2\};V}_{\beta;\boldsymbol{\epsilon}} & = \frac{\beta}{2}\,\mathcal{F}^{[0,0];V}_{\boldsymbol{\epsilon}}, \\ F^{\{-1\};V}_{\beta;\boldsymbol{\epsilon}} & = \frac{\beta}{2}\Big(1 - \frac{2}{\beta}\Big)\Big(\mathcal{F}^{[0,1];V}_{\boldsymbol{\epsilon}} + \ln\big(\tfrac{\beta}{2}\big)\Big) + \frac{\beta}{2}\ln\big(\tfrac{2\pi}{e}\big) - \ln \Gamma\big(\tfrac{\beta}{2}\big). \end{align} $$

The constant in the second line was not computed in [Reference Chekhov and EynardCE06]. To our knowledge, the absolute – including a $\beta $ -dependent, possibly g-dependent but otherwise V-independent constant – comparison between the coefficients $F_{\beta ;\boldsymbol {\epsilon }}^{\{k\};V}$ of the asymptotic expansion of the $\beta $ -ensembles and the invariants $\mathcal {F}^{[G,m]}$ for $(G,m) \neq (0,0),(0,1)$ produced by the topological recursion has not been performed in full generality. It is only known for $\beta = 2$ for all G in the one-cut regime; see [Reference MarchalMar17, Proposition 2.5].

When $\beta = 2$ , only $\mathcal {W}_{n;\boldsymbol {\epsilon }}^{[G]} = \mathcal {W}_{n;\boldsymbol {\epsilon }}^{[G,0]}$ and $\mathcal {F}_{\boldsymbol {\epsilon }}^{[G]} = \mathcal {F}^{[G,0]}_{\boldsymbol {\epsilon }}$ appear. These are the quantities defined by the Chekhov–Eynard–Orantin topological recursion [Reference Eynard and OrantinEO07], and we retrieve the usual asymptotic expansions

$$ \begin{align*} \mathcal{W}_{n;\boldsymbol{\epsilon}}(x_1,\ldots,x_n) & = \sum_{G \geq 0} N^{2 - 2G - n}\,\mathcal{W}_{n;\boldsymbol{\epsilon}}^{[G]}(x_1,\ldots,x_n), \\ \ln\bigg(\frac{Z_{N,\beta;\boldsymbol{\epsilon}}^{V;\mathsf{A}}}{Z_{N,\beta;\boldsymbol{\epsilon}}^{\tilde{V};\mathsf{A}}}\bigg) & = \sum_{G \geq 0} N^{2 - 2G}\big(\mathcal{F}^{[G];V}_{\boldsymbol{\epsilon}} - \mathcal{F}^{[G];\tilde{V}}_{\boldsymbol{\epsilon}}\big), \end{align*} $$

involving only powers of $\frac {1}{N}$ with parity $(-1)^n$ in the n-point correlators and powers of $\frac {1}{N^2}$ in the free energy.

1.6. Main results in the multi-cut regime: partition function

Let us come back to the initial model (1.1). We can always take $\mathsf {A} = \bigcup _{h = 0}^{g} \mathsf {A}_h \subseteq \mathsf {B}$ to be a small enlargement of the support $\mathsf {S}$ respecting the setup of § 1.4. It is indeed well known that the partition function $Z_{N,\beta }^{V;\mathsf {B}}$ can be replaced by $Z_{N,\beta }^{V;\mathsf {A}}$ up to exponentially small corrections when N is large (see [Reference Pastur and ShcherbinaPS11, Reference Borot and GuionnetBG11] for results in this direction, and we give a proof for completeness in § 3.1 below). The latter can be decomposed as a sum over all possible ways of distributing the $\lambda $ s between the segments $\mathsf {A}_h$ – namely,

(1.18) $$ \begin{align} Z_{N,\beta}^{V;\mathsf{A}} = \sum_{\substack{ N_0,\ldots,N_{g} \geq 0 \\ \sum_{h=0}^g N_h= N}} \frac{N!}{\prod_{h = 0}^{g} N_h!}\,Z_{N,\beta;\boldsymbol{N}/N}^{V;\mathsf{A}}, \end{align} $$

where we have denoted $N_0 = N - \sum _{h = 1}^{g} N_h$ the number of $\lambda $ s put in the segment $\mathsf {A}_0$ . So we can use our results for the model with fixed filling fractions to analyse the asymptotic behaviour of each term in the sum and then find the asymptotic expansion of the sum taking into account the interference of all contributions. This is carried out in Section 8.1.

Before stating the results, we need two ingredients. First, we let $\mathfrak {Z}_{N,\beta ;\boldsymbol {\epsilon }}^{V;\mathsf {A}}$ be the (truncated at an arbitrary order K) asymptotic series depending on a g-dimensional vector with positive entries, at least when its coefficients are defined:

(1.19) $$ \begin{align} \mathfrak{Z}_{N,\beta;\boldsymbol{\epsilon}}^{V;\mathsf{A}}= N^{\frac{\beta}{2}N + \varkappa}\exp\Big(\sum_{k = -2}^{K} N^{-k}\,F^{\{k\};V}_{\beta;\boldsymbol{\epsilon}} + O(N^{-(K + 1)})\Big)\,. \end{align} $$

If we substitute $\boldsymbol {\epsilon } = \boldsymbol {N}/N$ as in Theorem 1.4, it gives the asymptotic expansion of the partition function of the fixed filling fractions model with unordered eigenvalues, and we recall that $F^{\{k\};V}_{\beta ;\boldsymbol {\epsilon }}$ exists as a smooth function of $\boldsymbol {\epsilon }$ in some non-empty open set. We shall denote $(F^{\{k\};V}_{\beta ;\boldsymbol {\epsilon }})^{(j)}$ the tensor of j-th derivatives with respect to $\boldsymbol {\epsilon }$ .

Second, we introduce the Siegel Theta function with characteristics $\boldsymbol {\mu },\boldsymbol {\nu } \in \mathbb {C}^{g}$ . If $\boldsymbol {\tau }$ is a symmetric $g \times g$ matrix of complex numbers such that $\mathrm {Im}\,\boldsymbol {\tau }> 0$ , the Siegel Theta function is the entire function of $\boldsymbol {v} \in \mathbb {C}^{g}$ defined by the exponentially fast converging series

(1.20) $$ \begin{align} \vartheta\!\left[\begin{array}{@{\hspace{-0.02cm}}l@{\hspace{-0.02cm}}} \boldsymbol{\mu} \\ \boldsymbol{\nu} \end{array}\right]\!\!(\boldsymbol{v}|\boldsymbol{\tau}) = \sum_{\boldsymbol{m} \in \mathbb{Z}^{g}} \exp\Big(\mathrm{i}\pi (\boldsymbol{m} + \boldsymbol{\mu})\cdot\boldsymbol{\tau}\cdot(\boldsymbol{m} + \boldsymbol{\mu}) + 2\mathrm{i}\pi(\boldsymbol{v} + \boldsymbol{\nu})\cdot(\boldsymbol{m} + \boldsymbol{\mu})\Big). \end{align} $$

Among its essential properties, we mention the following:

• ○ for any characteristics $\boldsymbol {\mu },\boldsymbol {\nu }$ , it satisfies the diffusion-like equation $4\mathrm{i}\pi \partial _{\tau _{h,h'}}\vartheta = \partial _{v_h}\partial _{v_{h'}}\vartheta $ .

• ○ it is a quasi-periodic function with lattice $\mathbb {Z}^{g} \oplus \boldsymbol {\tau }(\mathbb {Z}^{g})$ : for any $\boldsymbol {m}_0,\boldsymbol {n}_0 \in \mathbb {Z}^{g}$ ,

  $$ \begin{align*}\vartheta\!\left[\begin{array}{@{\hspace{-0.02cm}}l@{\hspace{-0.02cm}}} \boldsymbol{\mu} \\ \boldsymbol{\nu} \end{array}\right]\!\!(\boldsymbol{v} + \boldsymbol{m}_0 + \boldsymbol{\tau}\cdot\boldsymbol{n}_0|\boldsymbol{\tau}) = \exp\big(2\mathrm{i}\pi\boldsymbol{m}_0\cdot\boldsymbol{\mu} - 2\mathrm{i}\pi \boldsymbol{n_0}\cdot(\boldsymbol{v} + \boldsymbol{\nu}) - \mathrm{i}\pi\boldsymbol{n}_0\cdot\boldsymbol{\tau}\cdot\boldsymbol{n}_0\big)\,\vartheta\!\left[\begin{array}{@{\hspace{-0.02cm}}l@{\hspace{-0.02cm}}} \boldsymbol{\mu} \\ \boldsymbol{\nu} \end{array}\right]\!\!(\boldsymbol{v}|\boldsymbol{\tau}). \end{align*} $$

• ○ it has a nice transformation law under $\boldsymbol {\tau } \rightarrow (\boldsymbol {A\tau } + \boldsymbol {B})(\boldsymbol {C\tau } + \boldsymbol {D})^{-1}$ , where $\boldsymbol {A},\boldsymbol {B},\boldsymbol {C},\boldsymbol {D}$ are the $g \times g$ blocks of a $2g \times 2g$ symplectic matrix [Reference MumfordMum84].

• ○ when $\boldsymbol {\tau }$ is the matrix of periods of a genus g Riemann surface, it satisfies the Fay identity [Reference FayFay70].

We define the gradient operator $\nabla _{\boldsymbol {v}}$ acting on the variable $\boldsymbol {v}$ of this function. For instance, the diffusion equation takes the form $4\mathrm{i}\pi \partial _{\boldsymbol {\tau }}\vartheta = \nabla ^{\otimes 2}_{\boldsymbol {v}}\vartheta $ .

Theorem 1.5. Assume Hypotheses 1.1 and 1.3. Let $\boldsymbol {\epsilon }_{\star } = (\mu _\mathrm{{eq}}^V[\mathsf {S}_h])_{1 \leq h \leq g}$ – we shall replace all indices $\boldsymbol {\epsilon }$ by $\star $ in our notations to indicate a specialisation at $\boldsymbol {\epsilon } = \boldsymbol {\epsilon }_\star $ . Then, the partition function has an asymptotic expansion of the form, with $\mathsf {C}=\mathsf {B}$ or $\mathsf {A}$ , for any $K \geq -2$ ,

(1.21) $$ \begin{align} Z_{N,\beta}^{V;\mathsf{C}} = \mathfrak{Z}_{N,\beta;\star}^{V;\mathsf{A}}\left\{\Big(\sum_{k = 0}^{K} N^{-k}\,T_{\beta;\star}^{\{k\}}\big[\tfrac{\nabla_{\boldsymbol v}}{2\mathrm{i}\pi}\big]\Big)\vartheta\!\left[\begin{array}{@{\hspace{-0.03cm}}c@{\hspace{-0.03cm}}} -N\boldsymbol{\epsilon}_{\star}\, \\ \boldsymbol{0} \end{array}\right]\!\!(\boldsymbol{v}_{\beta;\star}|\boldsymbol{\tau}_{\beta;\star}) + O(N^{-(K + 1)})\right\}. \end{align} $$

In this expression, $\mathfrak {Z}_{N,\beta ;\star }^{V;\mathsf {A}}$ is the asymptotic series defined in Equation (1.19) and evaluated at $\boldsymbol {\epsilon } = \boldsymbol {\epsilon }_{\star }$ . If $\boldsymbol {X}$ is a vector with g components, we set $T^{\{0\}}_{\beta ;\boldsymbol {\epsilon }}[\boldsymbol {X}] = 1$ , and for $k \geq 1$ ,

(1.22) $$ \begin{align} T_{\beta;\boldsymbol{\epsilon}}^{\{k\}}[\boldsymbol{X}] = \sum_{r = 1}^{k} \frac{1}{r!} \sum_{\substack{k_1,\ldots,k_r \geq -2 \\ j_1,\ldots,j_r> 0 \\ k_i + j_i > 0 \\ \sum_{i = 1}^{r} k_i + j_i = k}} \Big(\bigotimes_{i = 1}^{r} \frac{(F_{\beta;\boldsymbol{\epsilon}}^{\{k_i\};V})^{(j_i)}}{j_i!}\Big)\cdot\boldsymbol{X}^{\otimes(\sum_{i = 1}^r j_i)}, \end{align} $$

where $\cdot $ denotes the standard scalar product on the tensor space. We have also introduced

$$ \begin{align*}\boldsymbol{v}_{\beta;\star} = \frac{(F^{\{-1\};V}_{\beta;\star})'}{2\mathrm{i}\pi},\qquad \boldsymbol{\tau}_{\beta;\star} = \frac{(F^{\{-2\};V}_{\beta;\star})"}{2{\mathrm{i}\pi}}. \end{align*} $$

Being more explicit but less compact, we may rewrite

(1.23) $$ \begin{align} \nonumber \quad T_{\beta;\star}^{\{k\}}\big[\tfrac{\nabla_{\boldsymbol{v}}}{2\mathrm{i}\pi}\big]\vartheta\!\left[\begin{array}{@{\hspace{-0.03cm}}c@{\hspace{-0.03cm}}} -N\boldsymbol{\epsilon}_{\star}\, \\ \boldsymbol{0} \end{array}\right]\!\!(\boldsymbol{v}_{\beta;\star}|\boldsymbol{\tau}_{\beta;\star}) & = \sum_{r = 1}^{k} \frac{1}{r!} \sum_{\substack{k_1,\ldots,k_r \geq -2 \\ j_1,\ldots,j_r> 0 \\ k_i + j_i > 0 \\ \sum_{i = 1}^{r} k_i + j_i = k}}\!\!\!\! \Big(\bigotimes_{i = 1}^{r} \frac{(F_{\beta;\star}^{\{k_i\};V})^{(j_i)}}{j_i!}\Big) \\ & \quad \cdot \Big(\sum_{\boldsymbol{m} \in \mathbb{Z}^{g}}(\boldsymbol{m} - N\boldsymbol{\epsilon}_{\star})^{\otimes(\sum_{i = 1}^r j_i)}\,e^{\mathrm{i}\pi \cdot\boldsymbol{\tau_{\beta;\star}}\cdot(\boldsymbol{m} - N\boldsymbol{\epsilon}_{\star})^{\otimes 2} + 2\mathrm{i}\pi \boldsymbol{v}_{\beta;\star}\cdot(\boldsymbol{m} - N\boldsymbol{\epsilon}_{\star})}\Big). \end{align} $$

For $\beta = 2$ , this result has been derived heuristically to leading order in [Reference Bonnet, David and EynardBDE00] and to all orders in [Reference EynardEyn09]. These heuristic arguments can be extended straightforwardly to all values of $\beta $ ; see, for example, [Reference BorotBor11]. Our work justifies their heuristic argument. To prove this result, we exploit the Dyson–Schwinger equations for the $\beta $ -ensemble with fixed filling fractions taking advantage of a rough control on the large N behaviour of the correlators. The result of Theorem 1.5 has been derived up to $o(1)$ by Shcherbina [Reference ShcherbinaShc12] for real-analytic potentials, with different techniques, based on the representation of $\prod _{h < h'} \prod _{i,j} |\lambda _{h,i} - \lambda _{h',j}|^{\beta }$ , which is the exponential of a quadratic statistic, as expectation value of a linear statistics coupled to a Brownian motion. The rough a priori controls on the correlators do not allow at present the description of the $o(1)$ by such methods. The results in [Reference ShcherbinaShc12] were also written in a different form: $F^{\{0\};V}_{\beta ;\boldsymbol {\epsilon }}$ appearing in $\mathfrak {Z}$ was identified with a combination of Fredholm determinants (see also the physics paper [Reference Wiegmann and ZabrodinWZ06]), while this representation does not come naturally in our approach. Also, the steps undertaken in Section 8 where we replace the sum over nonnegative integers such that $N_0 + \cdots + N_g = N$ in Equation (1.18), by a sum over $\boldsymbol {N} \in \mathbb {Z}^{g}$ , thus reconstructing the Siegel Theta function, was not performed in [Reference ShcherbinaShc12].

The $2\mathrm{i}\pi $ appears because we used the standard definition of the Siegel Theta function,and should not hide the fact that all terms in Equation (1.23) are real-valued. Here, the matrix

(1.24) $$ \begin{align} \boldsymbol{\tau}_{\beta;\star} = \frac{\mathrm{Hessian}(F^{\{-2\};V}_{\beta;\boldsymbol{\epsilon}})\big|_{\boldsymbol{\epsilon} = \boldsymbol{\epsilon}_{\star}}}{2\mathrm{i}\pi} \end{align} $$

involved in the Theta function has purely imaginary entries, and $\mathrm {Im}\,\boldsymbol {\tau }_{\beta ;\star }$ is definite positive according to Theorem 1.4; hence, the Theta function in the right-hand side makes sense. Notice also that for it is $\mathbb {Z}^g$ -periodic with respect to $\boldsymbol {\mu }$ ; hence, we can replace $-N\boldsymbol {\epsilon }_{\star }$ by $-N\boldsymbol {\epsilon }_{\star } + \lfloor N\boldsymbol {\epsilon }_{\star } \rfloor $ , and this is responsible for modulations in the asymptotic expansion, and thus breakdown of the $\frac {1}{N}$ expansion. Still, the model has ‘subsequential’ asymptotic expansions in $\frac {1}{N}$ . For instance, for an even potential with two cuts ( $g = 1$ ) model, we have $\epsilon _{\star } = \frac {1}{2}$ , so $(-N\epsilon _{\star }\,\,\mathrm{mod}\,\,\mathbb {Z})$ appearing as characteristic in the Theta function only depends on the parity of N, and for each fixed parity, we get an asymptotic expansion in $\frac {1}{N}$ . In fact, having an even potential implies that the fixed-filling fraction model is invariant under $\epsilon \rightarrow 1 - \epsilon $ , so only the terms with even numbers $j_i$ of derivatives with respect to filling fractions contribute in $T^{\{k\}}_{\beta ;\boldsymbol {\epsilon }}$ . If, furthermore, $\beta = 2$ , only the $(F^{\{k\};V}_{\beta = 2;\star })^{(j)}$ with k even survive, and we deduce that the same is true for $T^{\{k\}}_{\beta =2;\star }$ , so that the logarithm of the partition function has an asymptotic expansion in $\frac {1}{N^2}$ for N odd and different asymptotic expansion in $\frac {1}{N^2}$ for N even (of course, up to the universal logarithmic corrections $\frac {\beta }{2}N\ln N + \varkappa \ln N$ ).

Let us give the two first orders of Equation (1.23):

$$ \begin{align*}T_{\beta;\star}^{\{1\}}[\boldsymbol{X}] = \frac{1}{6}\,(F_{\beta;\star}^{\{-2\};V})"'\cdot\boldsymbol{X}^{\otimes 3} + \frac{1}{2}\,(F_{\beta;\star}^{\{-1\};V})"\cdot\boldsymbol{X}^{\otimes 2} + (F_{\beta;\star}^{\{0\};V})'\cdot\boldsymbol{X}, \end{align*} $$

and:

$$ \begin{align*} T_{\beta;\star}^{\{2\}}[\boldsymbol{X}] & = \frac{1}{72}\,\big[(F_{\beta;\star}^{\{-2\};V})"'\big]^{\otimes 2}\cdot\boldsymbol{X}^{\otimes 6} + \frac{1}{12}\,\big[(F_{\beta;\star}^{\{-2\};V})"'\otimes (F_{\beta;\star}^{\{-1\};V})"\big]\cdot \boldsymbol{X}^{\otimes 5} \\ & \quad + \Big(\frac{1}{6}\,\big[(F_{\beta;\star}^{\{-2\};V})"'\otimes (F_{\beta;\star}^{\{0\};V})'\big] + \frac{1}{8}\,\big[(F_{\beta;\star}^{\{-1\};V})"\big]^{\otimes 2} + \frac{1}{24}\,(F_{\beta;\star}^{\{-2\};V})^{(4)}\Big)\cdot\boldsymbol{X}^{\otimes 4} \\ & \quad + \Big(\frac{1}{2}\,\big[(F_{\beta;\star}^{\{-1\};V})"\otimes(F_{\beta;\star}^{\{0\};V})'\big] + \frac{1}{6}\,(F_{\beta;\star}^{\{-1\};V})"'\Big)\cdot\boldsymbol{X}^{\otimes 3} \\ & \quad + \Big(\frac{1}{2}\,\big[(F_{\beta;\star}^{\{0\};V})'\big]^{\otimes 2} + \frac{1}{2}\,(F_{\beta;\star}^{\{0\};V})"\Big)\cdot\boldsymbol{X}^{\otimes 2} + (F_{\beta;\star}^{\{1\};V})'\cdot \boldsymbol{X}. \end{align*} $$

For $\beta = 2$ , unlike the one-cut regime where the asymptotic expansion was in $\frac {1}{N^2}$ up to constants independent of the potential, the multi-cut regime features an asymptotic expansion with nontrivial terms in powers of $\frac {1}{N}$ . For instance, we have a contribution at order $\frac {1}{N}$ of

$$ \begin{align*}T_{\beta = 2;\star}^{\{1\}}[\boldsymbol{X}] = \frac{1}{6}\,(F_{\beta = 2;\star}^{\{-2\};V})"'\cdot\boldsymbol{X}^{\otimes 3} + (F_{\beta = 2;\star}^{\{0\};V})'\cdot\boldsymbol{X}. \end{align*} $$

In a two-cuts regime ( $g = 1$ ), a sufficient condition for all terms of order $N^{-(2k + 1)}$ to vanish (again, up to integration constants already present in $\mathfrak {Z}$ ) is that $\epsilon _{\star } = \frac {1}{2}$ and $Z_{N,\beta = 2;\epsilon }^{V;\mathsf {A}} = Z_{N,\beta = 2;1 - \epsilon }^{V;\mathsf {A}}$ , for the same reasons that we mentioned for the case of an even potential with two cuts. In such a case, we have an expansion in powers of $\frac {1}{N^2}$ for the partition function, whose coefficients depend on the parity of N. In general, we also observe that $\boldsymbol {v}_{\beta = 2;\star } = \boldsymbol {0}$ (i.e., Thetanullwerten appear in the expansion).

Using the fact that the n-th correlator is the n-derivative of the free energy of the partition function for a perturbed potential or order $1/N$ , and our asymptotic results are uniform for small perturbations of this kind, it is pure algebra to derive from (1.5) an asymptotic expansion for the correlators $W_n$ for the initial model in the multi-cut regime. For $\beta = 2$ , the resulting expression can be found, for instance, in [Reference Borot and EynardBE11, Section 6.2] up to $O(\frac {1}{N})$ and a systematic diagrammatic for all orders is given in [Reference Borot and EynardBE12, Appendix A]. This can be straightforwardly extended to the $\beta \neq 2$ case simply by including half-integer genera g (in our conventions, k not having fixed parity).

1.7. Comments relative to the geometry of the spectral curve

We now stress facts from the theory of the topological recursion [Reference Chekhov and EynardCE06, Reference Eynard and OrantinEO07] which are relevant in the present case – for further details on the geometry compact Riemann surfaces, see, for instance, [Reference EynardEyn18]. When V is a polynomial and $\boldsymbol {\epsilon }$ is close enough to $\boldsymbol {\epsilon }_{\star }$ , the density of the equilibrium measure can be analytically continued to a hyperelliptic curve of genus g, denoted $\mathcal {C}_{\boldsymbol {\epsilon }}$ (the spectral curve). Its equation is

(1.25) $$ \begin{align} y^2 = \prod_{h = 0}^{g} (x - \alpha^{-}_{\boldsymbol{\epsilon},h})(x - \alpha^{+}_{\boldsymbol{\epsilon},h}), \end{align} $$

and $\mathcal {C}_{\boldsymbol {\epsilon }}$ is the compactification of the locus of such $(x,y)$ obtained by adding the two points at $\infty $ , where $y \sim x^{g +1}$ (first sheet) and $y \sim -x^{g+1}$ (second sheet). Let $\mathcal {A}_h$ be the cycle in $\mathcal {C}_{\boldsymbol {\epsilon }}$ surrounding $\mathsf {A}_{\boldsymbol {\epsilon },h} = [\alpha ^{-}_{\boldsymbol {\epsilon },h},\alpha ^{+}_{\boldsymbol {\epsilon },h}]$ . The family $\boldsymbol {\mathcal {A}} = (\mathcal {A}_h)_{1 \leq h \leq g}$ can be completed by a family of cycles $\boldsymbol {\mathcal {B}}$ so that $(\boldsymbol {\mathcal {A}},\boldsymbol {\mathcal {B}})$ is a symplectic basis of homology of $\mathcal {C}_{\boldsymbol {\epsilon }}$ . More precisely, the cycle $\mathcal {B}_h$ travels from $\alpha _{\boldsymbol {\epsilon },h}^{-}$ to $\alpha _{\boldsymbol {\epsilon },h - 1}^+$ in the second sheet and $\alpha _{\boldsymbol {\epsilon },h - 1}^+$ to $\alpha _{\boldsymbol {\epsilon },h}^-$ in the first sheet. The correlators $W_{n;\boldsymbol {\epsilon }}^{[G,K]}$ are meromorphic functions on $\mathcal {C}^n_{\boldsymbol {\epsilon }}$ , computed recursively by a residue formula on $\mathcal {C}_{\boldsymbol {\epsilon }}$ .

In particular, the analytic continuation of

(1.26) $$ \begin{align} \bigg(\frac{\beta}{2} W_{2;\boldsymbol{\epsilon}}^{\{0\}}(x_1,x_2) + \frac{1}{(x_1 - x_2)^2}\bigg)\mathrm{d} x_1\mathrm{d} x_2 = \bigg(\mathcal{W}_{2;\boldsymbol{\epsilon}}^{[0,0]}(x_1,x_2) + \frac{1}{(x_1 - x_2)^2}\bigg)\mathrm{d} x_1\mathrm{d} x_2 \end{align} $$

is the unique meromorphic bidifferential, denoted $\Omega $ , on $\mathcal {C}_{\boldsymbol {\epsilon }}$ , which has vanishing $\boldsymbol {\mathcal {A}}$ -periods and has for only singularity a double pole at coinciding point with leading coefficient $1$ and without residue. This $\Omega $ plays an important role for the geometry of the spectral curve and is called fundamental bidifferential of the second kind. It sometimes appears under the name of ‘Bergman kernel’, although it does not coincide with (but it is related to) the kernel introduced by Bergman in [Reference Bergman and SchifferBS53]. It can be explicitly computed by the formula

(1.27) $$ \begin{align} \Omega(z_1,z_2) = \mathrm{d}_{z_1} \mathrm{d}_{z_2} \ln \theta\Big(\int_{z_1}^{z_2} \boldsymbol{\varpi}\mathrm{d} x + \mathbf{c}\,\Big|\,\boldsymbol{\tau}^{\mathcal{C}_{\boldsymbol{\epsilon}}}\Big), \end{align} $$

where

• ○ $\theta = \vartheta \big [\begin {smallmatrix} \mathbf {0} \\ \mathbf {0} \end {smallmatrix}\big ]$ is the Riemann Theta function.

• ○ $\boldsymbol {\varpi }(z)\mathrm {d} x(z)$ is the basis of holomorphic one-forms dual to the $\boldsymbol {\mathcal {A}}$ -cycles – that is, characterised by

  (1.28)

  

• ○ $\boldsymbol {\tau }^{\mathcal {C}_{\boldsymbol {\epsilon }}}$ is the Riemann matrix of periods of the spectral curve $\mathcal {C}_{\boldsymbol {\epsilon }}$ :

  

• ○ $\mathbf {c} = \frac {1}{2}(\mathbf {r} + \boldsymbol {\tau}^{\mathcal{C}_{\boldsymbol{\epsilon}}}(\mathbf {s}))$ with $\mathbf {r},\mathbf {s} \in \mathbb {Z}^{g}$ such that $\mathbf {r} \cdot \mathbf {s}$ is odd, is a nonsingular characteristic for the Theta function (i.e., such that $\theta \big (\int _{z_1}^{z_2} \boldsymbol {\varpi }\mathrm {d} x + \mathbf {c}\,\big |\,\boldsymbol {\tau}^{\mathcal{C}_{\boldsymbol{\epsilon}}}\big )$ is not identically $0$ when $z_1,z_2 \in \mathcal {C}_{\boldsymbol {\epsilon }}$ ). Such a $\mathbf {c}$ exists and the result then does not depend on which such $\mathbf {c}$ is chosen.

It is a property of the topological recursion that the derivatives of $F_{\beta ;\boldsymbol {\epsilon }}^{\{k\};V}$ can be computed as $\boldsymbol {\mathcal {B}}$ -cycle integrals of the correlators:

(1.29) $$ \begin{align} (F_{\beta;\boldsymbol{\epsilon}}^{\{k\};V})^{(j)} = \Big(\frac{\beta}{2}\Big)^{j} \oint_{\boldsymbol{\mathcal{B}}} \mathrm{d} \xi_1 \cdots \oint_{\boldsymbol{\mathcal{B}}} \mathrm{d} \xi_j\,W_{j;\boldsymbol{\epsilon}}^{\{k + j\}}(\xi_1,\ldots,\xi_j). \end{align} $$

This relation extends as well to derivatives of correlators:

$$ \begin{align*}\big(W_{n;\boldsymbol{\epsilon}}^{\{k\}}(x_1,\ldots,x_n)\big)^{(j)} = \Big(\frac{\beta}{2}\Big)^{j} \oint_{\boldsymbol{\mathcal{B}}} \mathrm{d} \xi_1 \cdots \oint_{\boldsymbol{\mathcal{B}}} \mathrm{d} \xi_{j}\,W_{n + j;\boldsymbol{\epsilon}}^{\{k + j\}}(x_1,\ldots,x_k,\xi_1,\ldots,\xi_{j}), \end{align*} $$

where it is understood that we differentiate keeping x fixed. In particular,

(1.30) $$ \begin{align} \big(W_{1;\boldsymbol{\epsilon}}^{\{-1\}}(x)\big)' \mathrm{d} x = 2\mathrm{i}\pi\,\boldsymbol{\varpi}(x) \mathrm{d} x = \oint_{\boldsymbol{\mathcal{B}}} \Omega(x,\bullet) = \frac{\beta}{2} \oint_{\boldsymbol{\mathcal{B}}} \mathrm{d} \xi\,W_{2;\boldsymbol{\epsilon}}^{\{0\}}(x,\xi). \end{align} $$

Besides, the matrix to use in the Theta function appearing in Theorem 1.5 is

$$ \begin{align*}\tau_{\beta;\star} = \frac{\beta}{2}\, \boldsymbol{\tau}^{\mathcal{C}_{\boldsymbol{\epsilon}}}. \end{align*} $$

This simple dependence in $\beta $ of $W_{2;\boldsymbol {\epsilon }}^{\{0\}}$ can be traced back to the fact that, as a consequence of the Dyson–Schwinger equations, we have

and this equation (together with the properties of the analytic continuation of $W_{2;\boldsymbol {\epsilon }}^{\{0\}}$ on $\mathcal {C}_{\boldsymbol {\epsilon }}$ and the constraint of vanishing $\boldsymbol {\mathcal {A}}$ -periods) fully characterises $W_{2;\boldsymbol {\epsilon }}^{\{0\}}$ .

This relation has a long history and follows from the identification of $F^{\{-2\};V}_{\beta ;\boldsymbol {\epsilon }} = \frac {\beta }{2} \mathcal {F}^{[0,0];V}_{\boldsymbol {\epsilon }}$ (cf. Equation (1.17)) with the prepotential of the Hurwitz space associated to the family of curves (1.25) – considered as a Frobenius manifold – computed by Dubrovin [Reference DubrovinDub91], as well as with the tau function of the Whitham hierarchy as shown by Krichever [Reference KricheverKri92]. A derivation in the context of matrix model is, for instance, given in [Reference Chekhov and MironovCM02]. Although a priori differentiability of $F^{\{-2\};V}_{\beta ;\boldsymbol {\epsilon }}$ is not justified in [Reference Chekhov and MironovCM02], it is guaranteed by our results of Section A.2.

Equation (1.29) at $\boldsymbol {\epsilon } = \boldsymbol {\epsilon }_{\star }$ can be used to compute $T_{\beta ;\star }^{\{k\}}[\boldsymbol {X}]$ appearing in Equation (1.22). The derivation with respect to $\boldsymbol {\epsilon }$ is not a natural operation in the initial model when N is finite since $N\epsilon _h$ are forced to be integers in Equation (1.10). Yet we show that the coefficients of expansion themselves are smooth functions of $\boldsymbol {\epsilon }$ , and thus, $\partial _{\boldsymbol {\epsilon }}$ makes sense.

1.8. Central limit theorems for fluctuations and their breakdown

In Section 8.2, we describe the fluctuation of the number of particles $N_h$ in each segment $\mathsf {A}_h$ : when $N \rightarrow \infty $ , its law is approximated by the law of a Gaussian conditioned to live in a shifted integer lattice. The shift of the lattice oscillates with N by an amount $\lfloor N\epsilon _{\star ,h} \rfloor $ . Note that since $N \epsilon _{\star ,h}$ is for general N not an integer, strictly speaking, one cannot say that it converges in law to a discrete Gaussian random variable. This is, however, true along subsequences of N in case $\epsilon _{\star ,h} = \mu _{\mathrm{eq}}^{V}(\mathsf {A}_h)$ is a rational number.

Theorem 1.6. Assume Hypotheses 1.1 and 1.3, and let $\boldsymbol {N} = (N_1,\ldots ,N_g)$ be the vector of filling fractions as above. If $\boldsymbol {P}$ is a g-tuple of integers depending on N and such that $\boldsymbol {P} - N\boldsymbol {\epsilon }_{\star } = o(N^{\frac {1}{3}})$ when $N \rightarrow \infty $ , we have

(1.31) $$ \begin{align} \mu_{N,\beta}^{V;\mathsf{A}}\big(\boldsymbol{N} = \boldsymbol{P}\big) \sim \frac{e^{\frac{1}{2}\,(F^{\{-2\}}_{\beta;\star})^{"}\cdot(\boldsymbol{P} - N\boldsymbol{\epsilon}_{\star})^{\otimes 2} + (F^{\{-1\}}_{\beta;\star})'\cdot(\boldsymbol{P} - N\boldsymbol{\epsilon}_{\star})}}{\vartheta\big[\begin{smallmatrix} -N\boldsymbol{\epsilon}_{\star}\\ \boldsymbol{0} \end{smallmatrix}\big](\boldsymbol{v}_{\beta;\star}|\boldsymbol{\tau}_{\beta;\star})}. \end{align} $$

In Section 8.3, we describe the fluctuations of linear statistics in the multi-cut regime.

Theorem 1.7. Assume Hypotheses 1.1 and 1.3. Let $\varphi $ be an analytic test function in a neighbourhood of $\mathsf {A}$ , and $s \in \mathbb {R}$ . We have when $N \rightarrow \infty $ ,

(1.32) $$ \begin{align} & \quad \nonumber \mu_{N,\beta}^{V;\mathsf{A}}\big(e^{\mathrm{i}s\big(\sum_{i = 1}^N \varphi(\lambda_i) - N\int_{\mathsf{S}} \varphi(\xi)\mathrm{d}\mu_{\mathrm{eq}}^{V}(\xi)\big)}\big) \\ & \mathop{\sim}_{N \rightarrow \infty} \exp\Big(\mathrm{i}s\,M_{\beta;\star}[\varphi] - \frac{s^2}{2}\,Q_{\beta;\star}[\varphi,\varphi]\Big)\,\frac{\vartheta\!\left[\begin{smallmatrix} -N\boldsymbol{\epsilon}_{\star} \\ \boldsymbol{0} \end{smallmatrix}\right]\!\big(\boldsymbol{v}_{\beta;\star} + \mathrm{i}s\,\boldsymbol{u}_{\beta;\star}[\varphi]\big|\boldsymbol{\tau}_{\beta;\star}\big)}{\vartheta\!\left[\begin{smallmatrix} -N\boldsymbol{\epsilon}_{\star} \\ \boldsymbol{0} \end{smallmatrix}\right]\!\big(\boldsymbol{v}_{\beta;\star}\big|\boldsymbol{\tau}_{\beta;\star}\big)}, \end{align} $$

where

We recall that the $\varpi _h(x)\mathrm {d} x$ are the holomorphic one-forms from Equations (1.28)–(1.30), while $W_{1;\boldsymbol {\epsilon }}^{\{0\}}$ and $W_{2;\boldsymbol {\epsilon }}^{\{0\}}$ appear in the asymptotic expansion of the correlators in the model with fixed filling fractions (Theorem 1.3), and here they must be specialised at $\boldsymbol {\epsilon } = \boldsymbol {\epsilon }_{\star }$ .

Remark 1.4. In particular, $\boldsymbol {u}_{\beta ;\star }$ is a linear map associating to a test function $\varphi $ a g-dimensional vector. When $\varphi $ is such that $\boldsymbol {u}_{\beta ;\star }[\varphi ] = 0$ , the Theta functions cancel out, and we deduce that the random variable

$$ \begin{align*}\Phi_N[\varphi] := \sum_{i = 1}^N \varphi(\lambda_i) - N\int_{\mathsf{S}} \varphi(\xi)\mathrm{d}\mu_{\mathrm{eq}}^{V}(\xi) \end{align*} $$

converges in law to a Gaussian random variable with mean $M_{\beta ;\star }[\varphi ]$ and covariance $Q_{\beta ;\star }[\varphi ,\varphi ]$ . We remark that we have the alternative formula from (8.10):

$$ \begin{align*}\boldsymbol{u}_{\beta;\star}[\varphi] = \Big(\frac{1}{2\mathrm{i}\pi} \partial_{{\epsilon}_h}\int_{\mathsf{S}} \varphi(\xi)\,\mathrm{d}\mu_{\mathrm{eq};\boldsymbol{\epsilon}}^V(\xi)\Big)_{1 \leq h \leq g}\Big|_{\boldsymbol{\epsilon} = \boldsymbol{\epsilon}_{\star}},\end{align*} $$

showing that $\boldsymbol {u}_{\beta ;\star }[\varphi ] $ vanishes when $ \boldsymbol {\epsilon }_{\star }$ is a critical point of $\int _{\mathsf {S}} \varphi (\xi )\,\mathrm {d}\mu _{\mathrm{eq};\boldsymbol {\epsilon }}^V(\xi )$ . Even though our results are obtained for analytic potentials and test functions, this condition clearly makes sense with less regularity. In fact, it is possible to generalise our results and techniques to consider sufficiently smooth potential and test functions instead of analytic ones. We refer the interested reader to [Reference GuionnetG19, Sections 4 and 6] to such a generalisation in the one-cut case.

When $\boldsymbol {u}_{\beta ;\star }[\varphi ] \neq 0$ , the central limit theorem does not hold anymore. Instead, from the shape of the right-hand side, $\Phi _N[\varphi ]$ is approximated when $N \rightarrow \infty $ by the sum of two independent random variables: the first one is a Gaussian random variable with mean $M_{\beta ;\star }[\varphi ]$ and covariance $Q_{\beta ;\star }[\varphi ,\varphi ]$ , and the second one is the scalar product with $2\mathrm{i}\pi \boldsymbol {u}_{\beta ;\star }[\varphi ]$ (which is a vector in $\mathbb {R}^g$ when $\varphi $ is real-valued) of a random Gaussian vector conditioned to live on the lattice $-\lfloor N\boldsymbol {\epsilon }_{\star } \rfloor + \mathbb {Z}^{g}$ . This also displays N-dependent oscillations. These oscillations can be interpreted in physical terms from tunnelling of particles between different segments. One sees, indeed, than moving a single $\lambda _i$ from $\mathsf {A}_h$ to $\mathsf {A}_{h'}$ changes $\Phi _N[\varphi ]$ by a quantity of order $1$ , which is already the typical order of fluctuation of linear statistics when filling fractions are fixed.

The next term in the asymptotic expansion of the left-hand side of (1.32) is of relative order $O(\frac {1}{N})$ , which therefore gives the speed of convergence of the associated linear statistics of the empirical measure.

1.9. Asymptotic expansion of kernels and correlators

Once the result on large N expansion of the partition function is obtained, we can easily infer the asymptotic expansion of the correlators and the kernels by perturbing the potential by terms of order $\frac {1}{N}$ , maybe complex-valued, as allowed by Hypothesis 1.3.

1.9.1. Leading behaviour of the correlators

Although we could write down the expansion for the correlators as a corollary of Theorem 1.5, we bound ourselves to point out their leading behaviour. Whereas $W_n$ behaves as $O(N^{2 - n})$ in the one-cut regime or in the model with fixed filling fractions, $W_n$ for $n \geq 3$ does not decay when N is large in a $(g + 1)$ -cut regime with $g \geq 1$ . More precisely, we have the following.

Theorem 1.8. Assume Hypothesis 1.1 and 1.3 and that the number of cuts $(g + 1)$ is greater or equal to $2$ . When $N \rightarrow \infty $ , we have, uniformly when $x_1,\ldots ,x_n$ belongs to any compact of $(\mathbb {C}\setminus \mathsf {A})^n$ ,

$$ \begin{align*}W_2(x_1,x_2) = W_{2;\star}^{\{0\}}(x_1,x_2) + \Big(\boldsymbol{\varpi}(x_1)\otimes\boldsymbol{\varpi}(x_2)\Big)\cdot\nabla_{\boldsymbol{v}}^{\otimes 2}\ln\vartheta\!\left[\begin{array}{@{\hspace{-0.03cm}}c@{\hspace{-0.03cm}}} -N\boldsymbol{\epsilon}_{\star}\, \\ \boldsymbol{0} \end{array}\right]\!\!\big(\boldsymbol{v}_{\beta;\star}\big|\boldsymbol{\tau}_{\beta;\star}\big) + o(1)\,, \end{align*} $$

and for any $n \geq 3$ ,

$$ \begin{align*}W_n(x_1,\ldots,x_n) = \Big(\bigotimes_{i = 1}^n \boldsymbol{\varpi}(x_i)\Big)\cdot\nabla_{\boldsymbol{v}}^{\otimes n}\ln\vartheta\!\left[\begin{array}{@{\hspace{-0.03cm}}c@{\hspace{-0.03cm}}} -N\boldsymbol{\epsilon_{\star}}\, \\ \boldsymbol{0} \end{array}\right]\!\!\big(\boldsymbol{v}_{\beta;\star}\big|\boldsymbol{\tau}_{\beta;\star}\big) + o(1)\,. \end{align*} $$

Integrating this result over $\boldsymbol {\mathcal {A}}$ -cycles provides the leading order behaviour of n-th order moments of the filling fractions $\boldsymbol {N}$ , and the result agrees with Theorem 1.6.

1.9.2. Kernels

We explain in § 6.3 that the following result concerning the kernel – defined in Equation (1.4) – is a consequence of Theorem 1.3:

Corollary 1.9. Assume Hypothesis 1.1 and 1.3. There exists $t> 0$ such that, for any sequence of $\boldsymbol {N} = (N_1,\ldots ,N_{g})$ such that $|\boldsymbol {N}/N - \boldsymbol {\epsilon }_{\star }|_1 < t$ , the n-point kernels in the model with fixed filling fractions have an asymptotic expansion when $N \rightarrow \infty $ of the form, for any $K \geq 0$ ,

(1.33) $$ \begin{align} \mathsf{K}_{n,\boldsymbol{c};\boldsymbol{\epsilon}}(x_1,\ldots,x_n) = \exp\bigg[\sum_{j = 1}^{n} Nc_j\big(\ln(x_j) + 2\mathrm{i}\pi \chi_j\big) + \sum_{k = -1}^{K} N^{-k}\Big(\sum_{r = 1}^{k + 2} \frac{1}{r!}\mathcal{L}_{\boldsymbol{x},\boldsymbol{c}}^{\otimes r}[W_{r;\boldsymbol{\epsilon}}^{\{k\}}]\Big) + O(N^{-(K + 1)})\bigg], \end{align} $$

where $\mathcal {L}_{\boldsymbol {x},\boldsymbol {c}}$ is the linear form

(1.34) $$ \begin{align} \mathcal{L}_{\boldsymbol{x},\boldsymbol{c}}[f] = \sum_{j = 1}^{n} c_j\int_{\infty}^{x_j} \check{f}(x)\mathrm{d} x,\qquad \mathrm{where}\,\,\check{f}(x) = f(x) + \frac{1}{x} \mathop{\,\mathrm Res\,}_{x = \infty} f(\xi)\mathrm{d} \xi. \end{align} $$

The error terms in this expansion are uniform for $x_1,\ldots ,x_n$ in any compact of $\mathbb {C}\setminus \mathsf {A}$ .

The $(r,k) = 1$ term in (1.33) depends on choices for the path of integration from $\infty $ to $x_j$ (the other terms do not and are also unaffected by the difference between f and $\check {f}$ in (1.34)), and $\chi _j \in \mathbb {Z}$ . These two features are a manifestation of the fact that the definition of the kernel depends on a choice of determination for the complex logarithm; resolving them by the choice of suitable determinations and domain of definition leads to specific integer values for $\chi _j$ . These subtleties are explained in details in § 6.3 and can be ignored if all $c_j \in \mathbb {Z}$ (in that case, the definition of the kernel does not depend on choices).

Hereafter, if $\gamma $ is a smooth path in $\mathbb {C}\setminus \mathsf {S}_{\boldsymbol {\epsilon }}$ , we set $\mathcal {L}_{\gamma } = \int _{\gamma }$ , and $\mathcal {L}_{\gamma }^{\otimes r}$ is given by

$$ \begin{align*}\mathcal{L}_{\gamma}^{\otimes r}[W_{r;\boldsymbol{\epsilon}}^{\{k\}}] = \int_{\gamma}\mathrm{d} x_1\cdots\int_{\gamma}\mathrm{d} x_r\,W_{r;\boldsymbol{\epsilon}}^{\{k\}}(x_1,\ldots,x_r).\end{align*} $$

A priori, the integrals in the right-hand side of Equation (1.33) depend on the relative homology class in $\mathbb {C}\setminus \mathsf {A}$ of paths between $\infty $ to $x_i$ . A basis of homology cycles in $\mathbb {C}\setminus \mathsf {A}$ is given by $\overline {\boldsymbol {\mathcal {A}}} = (\mathcal {A}_h)_{0 \leq h \leq g}$ , and we deduce from Equation (1.11) that

(1.35)

Therefore, the only multivaluedness of the right-hand side comes from the first term $N \mathcal {L}_{\boldsymbol {x},\mathbf {c}}[W_{1;\boldsymbol {\epsilon }}^{\{-1\}}]$ , and given Equation (1.35) and observing that $N_h = N\epsilon _h$ are integers, we see that it exactly reproduces the monodromies of the kernels depending on $c_j$ .

We now come to the multi-cut regime of the initial model. If $\boldsymbol {X}$ is a vector with g components, and $\mathcal {L}$ is a linear form on the space of holomorphic functions on $\mathbb {C}\setminus \mathsf {S}_{\boldsymbol {\epsilon }}$ , let us define

$$ \begin{align*}\tilde{T}^{\{k\}}_{\beta;\boldsymbol{\epsilon}}[\mathcal{L};\boldsymbol{X}] = \sum_{r = 1}^{k} \frac{1}{r!} \sum_{\substack{j_1,\ldots,j_r \geq 1 \\ k_1,\ldots,k_r \geq -2 \\ n_1,\ldots,n_r \geq 0 \\ k_i + j_i + n_i> 0 \\ \sum_{i = 1}^{r} k_i + j_i + n_i = k}} \Big(\bigotimes_{i = 1}^{r} \frac{\mathcal{L}^{\otimes n_i}[(W_{n_i;\boldsymbol{\epsilon}}^{\{k_i\}})^{(j_i)}]}{n_i!\,j_i!}\Big)\cdot\boldsymbol{X}^{\otimes(\sum_{i = 1}^r j_i)}, \end{align*} $$

where we took as convention $W_{n = 0;\boldsymbol {\epsilon }}^{\{k\}} = F_{\beta ;\boldsymbol {\epsilon }}^{\{k\}}$ and the derivatives are computed for fixed xs. Then, as a consequence of Theorem 1.5, we have the following.

Corollary 1.10. Assume Hypothesis 1.1 and 1.3. With the notations of Corollary 1.9, the n-point kernels have an asymptotic expansion, for any $K \geq 0$ ,

$$ \begin{align*} \mathsf{K}_{n,\mathbf{c}}(\boldsymbol{x}) = \mathsf{K}_{n,\mathbf{c};\star}(\boldsymbol{x}) \frac{\Big(\sum_{k = 0}^K N^{-k}\,\tilde{T}_{\beta;\star}^{\{k\}}\big[\mathcal{L}_{\boldsymbol{x},\boldsymbol{c}},\frac{\nabla_{\boldsymbol{v}}}{2{\rm i}\pi}\big]\Big)\vartheta\!\left[\begin{smallmatrix} -N\boldsymbol{\epsilon}_{\star} \\ \boldsymbol{0} \end{smallmatrix}\right]\!\big(\boldsymbol{v}_{\beta;\star} + \mathcal{L}_{\boldsymbol{x},\boldsymbol{c}}[\boldsymbol{\varpi}]\big|\boldsymbol{\tau}_{\beta;\star}\big)}{\Big(\sum_{k = 0}^K N^{-k}\,T_{\beta;\star}^{\{k\}}\big[\frac{\nabla_{\boldsymbol{v}}}{2{\rm i}\pi}\bigr]\Big)\vartheta\!\left[\begin{smallmatrix} -N\boldsymbol{\epsilon}_{\star} \\ \boldsymbol{0} \end{smallmatrix}\right]\!\big(\boldsymbol{v}_{\beta;\star}\big|\boldsymbol{\tau}_{\beta;\star}\big)}\big(1 + O(N^{-(K + 1)})\big). \end{align*} $$

The first factor comes from evaluation of the right-hand side of Equation (1.33) at $\boldsymbol {\epsilon } = \boldsymbol {\epsilon }_{\star }$ , ${\mathcal {L}}_{\boldsymbol {x},\boldsymbol {c}} = \sum _{j = 1}^{n} c_j\int _{\infty }^{x_j}$ and $\boldsymbol {\varpi }\mathrm{{d}} x$ is the basis of holomorphic one-forms.

A diagrammatic representation for the terms of such expansion was proposed in [Reference Borot and EynardBE12, Appendix A].

1.10. Strategy of the proof

The key idea of this article is to establish an asymptotic expansion for the partition functions of our models for fixed filling fractions:

(1.36) $$ \begin{align} \frac{N!\,Z_{N,\beta;\boldsymbol{N}/N}^{V;{\mathsf{A}}}}{\prod_{h = 0}^{g} N_h!}= N^{\frac{\beta}{2}N + {\varkappa}}\exp\Big(\sum_{k = -2}^{K} N^{-k}\,F^{\{k\};V}_{\beta;\boldsymbol{N}/N} + O(N^{-(K + 1)})\Big), \end{align} $$

for any $K \geq 0$ . Indeed, such an expansion allows to estimate the free energy of the original model $\ln Z_{N,\beta }^{V;\mathsf {A}}$ up to errors of order $O(N^{-K-1 + \delta })$ ; see (1.18) and Theorem 1.5. It also allows to analyse the asymptotic distribution of the filling fractions $\boldsymbol {\epsilon } = \boldsymbol {N}/N$ (see Theorem 1.6) since this distribution is given as the following ratio of partition functions:

(1.37) $$ \begin{align} \mu_{N,\beta}^{V;\mathsf{A}}\big(\boldsymbol{N} \big)= \frac{N!}{\prod_{h = 0}^g N_h!}\,\frac{Z_{N,\beta;\boldsymbol{N}/N}^{V;\mathsf{A}}}{Z_{N,\beta}^{V;\mathsf{A}}}\,. \end{align} $$

In particular, if (1.36) is known up to $o(1)$ , the leading behaviour of (1.37) when $N \rightarrow \infty $ can be computed. This analysis is detailed in Section 8.2.

To handle fluctuations of linear statistics, we use the well-known approach of considering the free energy for perturbations of order $\frac {1}{N}$ of the potential. In fact, if we denote by $\Phi _N[\varphi ] := \sum _{i = 1}^N \varphi (\lambda _i) - N\int _{\mathsf {S}} \varphi (\xi )\mathrm {d}\mu _{\mathrm{eq}}^{V}(\xi ) $ , as in Remark 1.4, we see that for any real number s,

$$ \begin{align*}\mu_{N,\beta}^{V;\mathsf{A}}\big[e^{s\Phi_{N}[\varphi]}\big] = e^{-sN\int_{\mathsf{S}} \varphi(\xi)\mathrm{d}\mu_{\mathrm{eq}}^{V}(\xi) } \frac{Z_{N,\beta}^{V-\frac{2s}{N\beta}\varphi;\mathsf{A}}}{Z_{N,\beta}^{V;\mathsf{A}}}\,.\end{align*} $$

Again, the expansion of the free energies up to $o(1)$ allows to derive the asymptotics of the Laplace transform of $\Phi _{N}[\varphi ]$ and hence the central limit theorem; see Section 8.3. Note in passing that another way to study these fluctuations is to first condition the law $\mu _{N,\beta }^{V;\mathsf {A}}$ by fixing its filling fractions to be equal to some $\boldsymbol {N}$ . Indeed, we can also recover the fluctuations of the linear statistics from those under the conditioned law (that can be deduced from the ratio of the partition functions of Theorem 1.4 and lead to classical central limit theorems with Gaussian limits), together with the fluctuations of the filling fractions. Then, one easily sees that the term $\boldsymbol {u}_{\beta ;\star }$ comes from the fluctuations of the filling fractions and more precisely from the difference of centerings $N(\int _{\mathsf {S}} \varphi (\xi )\mathrm {d}\mu _{\mathrm{eq}}^{V}(\xi )-\int _{\mathsf {S}} \varphi (\xi )\mathrm {d}\mu _{\mathrm{eq};\boldsymbol {N}/N}^{V}(\xi ))$ for varying $\boldsymbol {N}/N$ .

Therefore, the central result of this article is Theorem 1.5. To prove this theorem, we shall as in [Reference Borot and GuionnetBG11] interpolate between the partition functions we are interested in and explicitly computable reference partition functions. For the latter, we take a product of partition functions of one-cut models with Gaussian, Laguerre or Jacobi weight (depending on the nature of edges, soft or hard, of the equilibrium measure one wishes to match) that are evaluated as Selberg integrals. Such reference partition functions were already used in [Reference Borot and GuionnetBG11]. One important new element of the present analysis is the interpolation from a model with several cuts to independent one-cut models. This is realised by considering the s-dependent model

$$ \begin{align*} & \quad Z_{N,\beta;\boldsymbol{\epsilon}}^{V;\mathsf{A}}(s) \\[7pt] & = \! \int_{\prod_{h = 0}^{g} \mathsf{A}_h^{N_h}} \Bigg[\prod_{h = 0}^{g} \prod_{i = 1}^{N_h} \mathrm{d}\lambda_{h,i}\,e^{-N\frac{\beta}{2}V_h(\lambda_{h,i})}\Bigg] \Bigg[\prod_{0 \leq h < h' \leq g} \prod_{\substack{1 \leq i \leq N_h \\ 1 \leq i' \leq N_{h'}}}\!\! |\lambda_{h,i} - \lambda_{h',i'}|^{s\beta}\Bigg] \Bigg[\prod_{h = 0}^{g} \prod_{1 \leq i < j \leq N_{h}} \!\! |\lambda_{h,i} - \lambda_{h,j}|^{\beta}\Bigg] \end{align*} $$

for $s \in [0,1]$ . We choose to take the s-dependent potential $V_h(x) = T^s_h(x)$ on the h-segment

$$ \begin{align*}T^s_h(x) = V(x) - 2(1 - s)\sum_{h' \neq h} \int_{\mathsf{A}_{h'}} \mathrm{d}\mu_{\mathrm{eq};\boldsymbol{\epsilon}}^{V}(\xi)\,\ln|x - \xi|,\qquad \mathrm{for}\,\,x \in \mathsf{A}_{h}\,, \end{align*} $$

where V is the potential of the original model. This choice is such that the equilibrium measure associated with the model $Z_{N,\beta ;\boldsymbol {\epsilon }}^{T_{s};\mathsf {A}}(s) $ is the equilibrium measure of the original model; see Section 7.4. Moreover, $Z_{N,\beta ;\boldsymbol {\epsilon }}^{V;\mathsf {A}}=Z_{N,\beta ;\boldsymbol {\epsilon }}^{T_{1};\mathsf {A}}(1) $ , whereas $Z_{N,\beta ;\boldsymbol {\epsilon }}^{T_{0};\mathsf {A}}(0)$ is a product of models whose equilibrium measure has only one-cut (they are the restriction of the equilibrium measure of the original model to each of the connected pieces of its support), which we can compute by [Reference Borot and GuionnetBG11] (see Section 7.1). Interpolating along this family yields

(1.38) $$ \begin{align} \nonumber & \quad \ln\bigg(\frac{Z_{N,\beta;\boldsymbol{\epsilon}}^{V;\mathsf{A}}}{Z_{N,\beta;\boldsymbol{\epsilon}}^{T_0;\mathsf{A}}(s = 0)}\bigg) =\int_{0}^{1}\partial_{s}\ln Z_{N,\beta;\boldsymbol{\epsilon}}^{T_{s};\mathsf{A}}(s) \mathrm{d} s \\ \nonumber &= \beta \int_{0}^{1} \mathrm{d} s\mu_{N,\beta;\boldsymbol{\epsilon}}^{T_{s};\mathsf{A}}(s) \bigg[\sum_{0 \leq h < h' \leq g}\sum_{{\substack{1 \leq i \leq N_h \\ 1 \leq i' \leq N_{h'}}}}\ln|\lambda_{h,i} - \lambda_{h',i'}| -N\sum_{0 \leq h' \neq h \leq g}\sum_{i = 1}^{N_h} \int_{\mathsf S_{h'}} \ln |\lambda_{h,i}-x| \mathrm{d}\mu_{\mathrm{eq};\boldsymbol{\epsilon}}^{V}(x)\bigg] \\ \nonumber & = -N\beta \sum_{0 \leq h \neq h' \leq g} \oint_{\mathsf{A}_{h}}\oint_{\mathsf{A}_{h'}} \frac{\mathrm{d} x\,\mathrm{d} x'}{(2\mathrm{i}\pi)^2}\,\ln[(x - x')\mathrm{sgn}(h - h')]\,W_{1;\boldsymbol{\epsilon}}^{\{-1\}}(x)\bigg(\int_{0}^{1} \mathrm{d} s\,W_{1;\boldsymbol{\epsilon}}^{s}(x')\bigg) \\ &\quad + \sum_{0 \leq h' \neq h \leq g} \frac{\beta}{2} \oint_{\mathsf{A}_{h}}\oint_{\mathsf{A}_{h'}} \frac{\mathrm{d} x\,\mathrm{d} x'}{(2\mathrm{i}\pi)^2} \ln[(x - x')\mathrm{sgn}(h - h')]\bigg(\int_{0}^{1} \mathrm{d} s \big[W_{2;\boldsymbol{\epsilon}}^{s}(x,x') + W_{1;\boldsymbol{\epsilon}}^{s}(x)W_{1;\boldsymbol{\epsilon}}^{s}(x')\big]\bigg). \end{align} $$

It is important to note that in the first equality, the singularity of the logarithm is away from the range of integration as it involves variables in distinct segments, so we could express (1.38) in terms of analytic linear and quadratic statistics, which, in turn, can be expressed in terms of the correlators $W_{n;\boldsymbol {\epsilon }}^{s}$ of the model associated with $Z_{N,\beta ;\boldsymbol {\epsilon }}^{T_{s};\mathsf {A}}(s)$ . Lemma 7.5 gives the large N expansion of these correlators.

These expansions are based on the so-called Dyson–Schwinger equations (4.1); see also (7.36) for the correlators of the interpolating models. These equations are exact equations satisfied by the correlators for any fixed N and obtained simply by integration by parts. They are a priori not closed, but the idea is to show that they are asymptotically closed so that if we can show that the correlators have a large N expansion of topological type, their coefficients will satisfy a closed system of equations. The latter is based on the fact that coefficients beyond the leading order satisfy an inhomogeneous linear equation, with inhomogeneous term involving coefficients of lower order only. Hence, solving the linear equation allows to define uniquely and recursively all the coefficients in the expansion of the correlators. The linear equation is described by a linear operator, called the master-operator, that we denote $\mathcal K$ (see (5.6)) and which is the same for all orders. An inversion of this operator (continuously on some function space) precisely allows to solve the linear equation.

The central point of our approach is therefore to invert the operator $\mathcal K$ . In fact, the operator is not invertible but rather has a kernel of dimension at least g, where $(g + 1)$ is the number of cuts (i.e., connected components of the support of the equilibrium measure). However, its extension $\hat {\mathcal K}$ , where we also record the periods around the cuts, is invertible in an off-critical situation; see Section 5.2.3. Fixing the filling fractions exactly amounts to use the extended operator $\widehat {\mathcal {K}}$ instead of $\mathcal {K}$ , and this is why we first consider the model with fixed filling fractions. The invertibility of the extended operator indeed allows us not only to formally solve the Dyson–Schwinger equations but also to show the existence of this asymptotic expansion to all orders in $\frac {1}{N}$ . To this end, it is necessary to use a priori rough estimates on the correlators, which we obtain by classical methods of concentration of measure and large deviations; see Section 3. These estimates can be improved iteratively with the Dyson–Schwinger equations (see, for example, Section 5.3) to obtain optimal estimates and eventually reach the all-order asymptotic expansion. This bootstrap strategy was first introduced in [Reference Borot and GuionnetBG11] for the one-cut model. We detail these computations in the case where $s=1$ in Section 5. We also need to carry this out for the interpolating s-dependent model in order to have asymptotic expansions to insert in (1.38). In that case, the extended operator does not have an explicit inverse, but we can nevertheless show by Fredholm arguments that it is invertible. Then we indicate in Section 7 the modifications to take into account the previous bootstrap argument for $s\in [0,1]$ .

We stress again that we cannot use the inversion and bootstrap strategy in the Dyson–Schwinger equations for the correlators of the original model in the multi-cut regime because the relevant master operator is not invertible. This is the reason why we need the detour through the partition function with fixed filling fractions (via (1.36)), from which any desired expansion of the correlators of the original model can be obtained by looking at $\frac {1}{N}$ -perturbations of the potential.