1.1 OPUC preliminaries and formulation of main results

A major advance in the study of $M_n$ was achieved in [Reference Chhaibi, Madaule and NajnudelCMN18], who used the Orthogonal Polynomials on the Unit Circle (OPUC) representation of the C $\beta $ E measure due to [Reference Killip and NenciuKN04]; we refer to [Reference SimonSim04] for an encyclopedic account of the OPUC theory. Let $\left \{ \gamma _k \right \}$ be independent, complex, rotationally invariant random variables for which $|\gamma _k|^2 = \operatorname {Beta}(1, \beta (k+1)/2)$ – that is, with density on $[0,1]$ proportional to $(1-x)^{\beta (k+1)/2-1}$ . The Szegő recurrence is, for all $k \geq 0,$

(1.5)

where and are polynomials of degree at most k. Define in terms of these coefficients the Prüfer phases

(1.6)

where here and below, we take the principal branch of the logarithm with discontinuity along the negative real line. Then, may be identified as a continuous version of the logarithm of (see [Reference Chhaibi, Madaule and NajnudelCMN18, Lemma 2.3]). These polynomials can be used to give an effective representation for the characteristic polynomial $X_n$ by setting $\alpha $ to be a uniformly distributed element of the unit circle, independent of $\left \{ \gamma _k : k \geq 0 \right \},$ and setting for any $\theta \in {\mathbb R},$

(1.7)

See [Reference Chhaibi, Madaule and NajnudelCMN18, (2.2)] for details of the demonstration, based on [Reference Killip and NenciuKN04, Proposition B.2], that this indeed has the law of the characteristic polynomial of C $\beta $ E.

The polynomials satisfy the recurrence

(1.8)

We also recall the relative Prüfer phase [Reference Chhaibi, Madaule and NajnudelCMN18, Lemma 2.4] given by the recurrence

(1.9)

In law, is equal to

We will be interested in the extreme values of the fluctuations of real and imaginary parts of for which reason we will formulate our results in terms of the recurrence

(1.10)

where $\sigma $ is one of Then, for $\sigma =1$ , , while for $\sigma =i$ , . The entire analysis of $\varphi $ occurs for a fixed $\sigma ,$ and so we shall not display the dependence on $\sigma .$

Formulation of main results

We already stated our main result concerning the maximum of the real part of the logarithm of the characteristic polynomial. In this section, we describe the rest of our results.

We recall the following, with $m_n = \log n - \tfrac {3}{4}\log \log n$ as in (1.2).

We shall expand upon this result and show that, indeed, this maximum converges in distribution. Moreover, we shall show that the process of almost maxima converges. The following result, which complements Theorem 1.1, yields the convergence in law of the centered maxima of $\varphi _n$ .

Remark 1.5. By the distributional identity,

which follows from the conjugation invariance of the law of $\gamma _k$ and the symmetry $-2\Im \log (1-z) = 2 \Im \log (1-\overline {z})$ , the case of $\sigma =-i$ in Theorem 1.4 follows similarly to the case of $\sigma =i$ .

To describe the random shift, we need to introduce some machinery.

1.2 The derivative martingale

We will need the so-called derivative martingale. Define the random measure and its total mass

(1.11)

We equip the space of finite measures with the weak-* topology, and then we show that this measure converges almost surely. We recall (see (1.10)) that $\varphi _k(\cdot )$ coincides with either twice the real or imaginary parts of (depending on $\sigma $ ).

Theorem 1.6. For any and any $\beta> 0,$ there is an almost surely finite random variable $\mathscr {B}_\infty $ and an almost surely finite, nonatomic random measure $\mathscr {D}_\infty $ so that

Furthermore, for any $\epsilon> 0$ , there is a compact $K \subset (0,\infty )$ so that with

it holds that for any $k \in {\mathbb N}$ ,

(1.12)

This is shown in Section 9. We remark that $(\mathscr {B}_{2^j} : j \in {\mathbb N})$ is not in fact a martingale, but it is easily compared to a process $(\widehat {\mathscr {B}_{2^j}} : j \in {\mathbb N})$ which is a martingale (see Section 9 for details).

1.3 Sequential Poisson process approximation and extremal landscape

We introduce parameters $\{k_p : p \in {\mathbb N}\}$ which will be chosen large but independent of $n.$ These parameters will be taken large after n is sent to infinity. Moreover, they will be ordered in a decreasing fashion so that $k_j \gg k_{j+1}.$ We shall not attempt to find any quantitative dependence of how these parameters are sent to infinity. All parameters will be assumed to be larger than $1.$

We formulate several sequential extremal processes approximation for the processes of near maxima; these extremal processes will be indexed by $k_1$ . Divide the unit circle into consecutive arcs $\{ \widehat {I_{j,n}}\}$ by the formula that for any $j,n \in {\mathbb N},$

(1.13)

To avoid cumbersome notation, we suppress the n dependence in $\widehat {I_{j,n}}$ , writing $\widehat {I_j}$ instead, and we continue to do so in the forthcoming $D_{j,n},\theta _{j,n},\widehat {W}_{j,n}$ . Let ${\mathcal {D}}_{n/k_1}$ denote the collection of indices $j=1,2,\dots ,\lceil \tfrac {n}{k_1}\rceil .$ We let $\theta _j=\theta _{j,n}$ be the supremum of $\widehat {I_{j,n}}.$ Over each of these intervals, we define the process

(1.14)

This will serve as the decoration process, although we will not need to (and will not) prove their convergence as $k_1\to \infty $ .

We next define for all $j \in {\mathcal {D}}_{n/k_1}$ random variables

(1.15)

which is a local maximum, appropriately centered. We now define a random measure which we shall show is well-approximated by a Poisson process with random intensity. Define a measure on Borel subsets of

(1.16)

by

(1.17)

A central technical challenge will be to show that $\varphi _k$ and are essentially constant on the interval $\widehat {I_j}$ for $k \approx n/{k_1}$ , and that hence, it suffices to track both $\varphi _k$ and only at the point $\theta _j \in \widehat {I_j}.$ In this direction, it will be helpful to further decompose the local maximum $\widehat {W_j}.$ We define two new parameters $k_1^+$ and $\widehat {k}_1$ as functions of $k_1$ in such a way that $k_1^+/\widehat {k}_1, \widehat {k}_1/k_1 \to \infty ,$ specifically:

(1.18) $$ \begin{align} {k_1^+} = {k_1}\exp({(\log k_1)}^{(29/30)}) \quad\text{and}\quad {\widehat{k}_1} = {k_1}\exp({(\log k_1)}^{(19/20)}). \end{align} $$

We define $n_1 = \lfloor n/k_1\rfloor ,$ and we define $\widehat {n}_1$ and $n_1^+$ analogously. Define

(1.19)

Define for Borel subsets of

(1.20)

Note that the processes $\operatorname {Ex}_n$ and $\operatorname {Ext}_n$ are closely related; see the proof of Theorem 1.11. Our goal will be to approximate the processes $\operatorname {Ext}_n$ (and $\operatorname {Ex}_n$ ) by Poisson processes with random intensity.

Toward this end, recall that an important strategy used throughout the analysis of extrema of branching processes is effectively conditioning on the initial portion of the process, wherein the extrema gain a nontrivial correlation. We will do the same and condition on the first Verblunsky coefficients. We use the parameter $k_2,$ which we assume is a power of $2$ (to apply Theorem 1.6), to refer to how many Verblunsky coefficients on which we condition. We also use $(\mathscr {F}_{k}: k \in {\mathbb N}_0)$ to refer to the natural $\sigma $ -algebra generated by the sequence of Verblunsky coefficients $( \gamma _k : k \in {\mathbb N}_0).$

We will compare $\operatorname {Ext}_n$ to the $\mathscr {F}_{k_2}$ -conditional Poisson random measures , , with respective intensity measures

(1.21) $$ \begin{align} &\mathscr{D}_{k_2}(\theta)d\theta \times I(v)dv \times \mathfrak{p}_{k_1}(v,df), \quad \mathscr{D}_{k_2}(\theta)d\theta \times I'(v)dv \times \mathfrak{p}_{k_1}(v,df), \quad \nonumber\\ &\qquad \mathscr{D}_{k_2}(\theta)d\theta \times I"(v)dv \times \mathfrak{p}_{k_1}(v,df), \end{align} $$

where

Here and in many places below, we slightly abuse notation by using $\times $ to denote also products of transition kernels (i.e., semidirect products). Thus, $I'$ is defined over a slightly longer interval than I, and $I"$ is defined over a slightly shorter interval than I. The law $\mathfrak {p}_{k_1}(v,\cdot )$ is that of a random function on which is related to the exponential of the solution of a family of coupled diffusions $\mathfrak {U}^o_{T_+}(\theta )$ in an auxiliary time parameter (see (2.78) and (2.52)).

Similarly, the measure $\operatorname {Ex}_n$ will be approximated by a Poisson random measure with a random intensity on the same space. This intensity on will take the form of a product measure $\mathscr{D}_\infty \times \widehat{\mathfrak{p}_{k_1}}$ , where $(\widehat {\mathfrak {p}_{k_1}} :k_1 \in {\mathbb N})$ is a deterministic Radon measure on , which is constructed as follows. Let

(1.22)

be a map of to itself, and let

(1.23) $$ \begin{align} \widehat{\mathfrak{p}_{k_1}}(dv,df)\; \text{denote the push-forward of }I(v)dv\times \mathfrak{p}_{k_1}(v,df)\text{ by }\iota. \end{align} $$

We let be a Poisson random measure on with intensity $\mathscr{D}_\infty \times \widehat{\mathfrak{p}_{k_1}}$ . We may define similarly and .

To compare point processes on $\Gamma $ , we endow the latter with the distance

In terms of this, we define (compare with $d_1'$ from [Reference Chen and XiaCX11]) a Wasserstein distance on point configurations $\xi _1 = \sum _{i=1}^m \delta _{y_i}$ and $\xi _2 = \sum _{i=1}^n \delta _{z_i}$

with the minimum being the distance over all permutations of $\{1,2,\dots ,n\}.$ Finally, for two point processes $Q_1$ and $Q_2$ , we define the pseudometric

with the infimum over couplings $(\xi _1, \xi _2)$ in which $\xi _1 \sim Q_1$ and $\xi _2 \sim Q_2.$ Note that this pseudometric only depends on the laws of the point processes. These distances are somewhat unorthodox; we develop Poisson approximations using these distances in Appendix A, as well as some comparisons of these distances to other more standard metrics used in Poisson approximation.

To get a comparison between $\operatorname {Ext}_n$ and the point processes , it is necessary to restrict to the case that the maximum of the modulus of the decoration $|D_je^{\sqrt {4/\beta }V_j}|$ is sufficiently large. So, we set

(1.24)

In what follows, for any measure (or point process) $\mathbf {Q}$ on $\Gamma $ , we write $\iota \# \mathbf {Q}$ for the push forward under the transformation that preserves the first coordinate and applies $\iota $ of (1.22) to the last two. Note that under $\iota \#\mathbf {Q}$ , the second coordinate of the point process can always be recovered from the third. Our main result, which will imply Theorems 1.1 and 1.4, is the following convergence of marked processes. Here and in the sequel, for a point process and a set $B\subset \Gamma $ , we write for the restriction of to B (i.e., ).

Theorem 1.10. The restrictions of $\operatorname {Ext}_n^{k_1}$ and to $\Gamma _{k_7}$ satisfy

(1.25)

The same holds with or replacing in (1.25).

The meaning of the $\limsup $ is that the parameters are taken to infinity in order, with n followed by $k_1$ followed by $k_2.$

Similarly, to make a comparison between and $\operatorname {Ex}_n^{k_1},$ we will only make a comparison in which their second coordinate is in a compact set. Hence, we shall further restrict the space ${\Gamma }$ from (1.16) to

(1.26)

Theorems 1.10 and 1.6 lead directly to the following:

Theorem 1.11. For any $k_7> 0,$ the restrictions of the point processes to $\widehat {\Gamma }_{k_7}$ satisfy

(1.27)

The same holds with or replacing in (1.27)

Equipped with Theorem 1.11, we can now complete the proof of Theorems 1.1 and 1.4, assuming a technical estimate contained in Corollary 10.3.

Proof of Theorems 1.1 and 1.4.

We give first the details for Theorem 1.1 since its proof is more involved. Using (1.7), we have the representation for all $z \in {\mathbb C}$

where we recall $\alpha $ is uniformly distributed on the unit circle and independent of We begin by proving convergence. From Theorem 1.3, as $m_{n+1}-m_{n} \to 0$ as $n \to \infty ,$ the random variables

are tight. Let $R_*$ be any subsequential weak limit.

We recall that and hence on the unit circle, we have

Thus, for $|z|=1$ , we have the representation of the log-modulus of the characteristic polynomial

(1.28)

Then using (1.14), we can represent $R_{n+1}$ by

(1.29)

By (1.28), the log-modulus of the characteristic polynomial can only increase by a $\log 2$ over the reversed OPUC and so for any $k_7$ sufficiently large,

Let for all $x \in {\mathbb R}.$ Then, on the event $\max _j \widehat {W}_j>-k_7$ ,

We would like to apply Theorem 1.11 to establish the convergence of this statistic. For this, we need a Lipschitz bound on the mapping

(1.30)

with respect to the metric on point configurations. We note that there is n-dependence in the mapping (1.30) beyond the n dependence in the process $\operatorname {Ex}_n$ , which we would like to remove. It follows from the definition of the metric that (1.30) is $C(k_7)$ -Lipschitz. Moreover, the difference

Now the mapping has a Lipschitz constant with respect to the sup-norm that is bounded solely in terms of $k_7.$ Thus, if we define for each $k_1$ a random variable by

then from Theorem 1.11 and the monotonicity of

with the supremum over all $1$ -Lipschitz real-valued functions $\vartheta $ . It follows that

Recall that $R_*$ is any subsequential limit point of $\{R_n\},$ and hence, for any other subsequential limit $R_*',$

As $k_7$ is arbitrary, it follows that $\{R_n\}$ has a unique weak- $*$ limit point (i.e., it converges in law). We also observe that since Theorem 1.11 also applies to or , in place of , the same argument above holds if in the definition of $\mathcal {R}^{k_1}$ , we replace by either of these. Hence, if we define $\mathcal {R}^{k_1,'}$ and $\mathcal {R}^{k_1,"}$ by making that replacement, then for all $k_7$ ,

We next characterize the limit. For that, it is enough to evaluate for fixed $x\in {\mathbb R}$ the limit of the probability $\mathbb {P}(\mathcal {R}^{k_1}\leq x)$ as $k_1\to \infty $ . Recall (1.29). For any with , any $x \in {\mathbb R}$ and any introduce the Borel subset of

From the definition of ,

and by approximation of the step function, we have for all x,

$$\begin{align*}\lim_{k_1 \to \infty} | \mathbb{P}(\mathcal{R}^{k_1}\leq x)- \mathbb{P}(\mathcal{R}^{k_1,'}\leq x) | + | \mathbb{P}(\mathcal{R}^{k_1}\leq x)- \mathbb{P}(\mathcal{R}^{k_1,"}\leq x) |=0. \end{align*}$$

Using the fact that is Poisson of random intensity, the last probability can be written as , where

Let

From Corollary 10.3, we have that uniformly on compact sets of $\alpha $ and y and uniformly in $|v| \leq (\log k_1)^{17/18}$ ,

Note that due to the random phase in the definition of $\mathfrak {p}_{k_1}$ , the function is actually independent of , and we can write . Then, using the change of variables $v=w-\sqrt {\tfrac {\beta }{4}}x$ and setting $J=[(\log k_1^+)^{1/10}, (\log k_1^+)^{9/10}]$ ,

In particular, does not depend on , and we can omit it from the notation. Since x is fixed, we obtain for w in the stated interval that $(w-\sqrt {\tfrac {\beta }{4}}x)/w=1+o_{k_1}(1)$ . Hence, if we let $\mathcal {I}_{x}'$ and $\mathcal {I}_{x}"$ to denote $\mathcal {I}_{x}$ with $I',I"$ replacing I, respectively, we obtain

(1.31) $$ \begin{align} \mathcal{I}_{0}"(1+o_{k_1}(1))\leq e^{x \sqrt{2\beta} } \mathcal{I}_{x}\leq \mathcal{I}_{0}'(1+o_{k_1}(1)). \end{align} $$

Using that all of the distributions functions of $\mathcal {R}^{k_1}, \mathcal {R}^{k_1,'}, \mathcal {R}^{k_1,"}$ converge together in the limit, we conclude

where

(1.32) $$ \begin{align} A_\beta=\int_{J} w e^{\sqrt{2}w }F_{\beta}(w,-\log 2+\sqrt{\tfrac{4}{\beta}}v) dw \end{align} $$

is a constant that does depend on , and which takes values in $(0,\infty )$ due to tightness.

The proof of Theorem 1.4 is identical, except that instead of working with $R_n$ as in (1.29), we can work directly with $\varphi _n$ , and in the right-hand side of (1.29), one replaces the expression by $\log D_j(\theta )$ , resulting in a simplification of the proof.

Theorem 1.11 is a direct corollary of Theorem 1.10 and some estimates from Section 2 below.

Proof of Theorem 1.11.

Note first that from the data (θ _j , V _j , D _j e ^{4/βV _j} ), we can express the triple (θ _j , W^ _j , D _j ) by a continuous transformation of the second two coordinates using $\iota $ , viz. $ (\widehat {W}_j, D_j) =\iota ( V_j, D_je^{\sqrt {4/\beta }V_j}).$ Moreover, this maps $\Gamma _{k_7}$ to $\widehat {\Gamma }_{k_7}.$ Furthermore, the transformation $\iota $ is Lipschitz with some constant $L({k_7})$ when restricted to this set. Let and $Q = \operatorname {Ext}_n^{k_1} \cap \Gamma _{k_7}.$ Hence, for any $k_2$ ,

(1.33)

where $\widehat {\iota }(P)$ is a Poisson point process on $\widehat {\Gamma }_{k_7}$ with intensity $\mathscr{D}_{k_2}(\theta)d\theta \times \widehat{\mathfrak{p}_{k_1}}$ and where we recall that $\widehat {\mathfrak {p}_{k_1}}$ is the pushforward of $I(t)dt \times \mathfrak {p}_{k_1}(t,df)$ under $\iota $ , which is a Radon measure. The first term in (1.33) goes to $0$ from Theorem 1.10.

We now turn to the second term. Define for any two finite Borel measures on $\Gamma ,$

with the supremum over all g with both for all $x,y \in \Gamma $ and $|g(x)| \leq 1$ for all $x \in \Gamma .$ We need the following bound:

(1.34)

which is shown in Lemma 2.24 (in the notation of that lemma, it is $\mathcal {H}$ ). If $f : \widehat {\Gamma }_{k_7} \to {\mathbb R}$ is a $1$ -bounded, $1$ -Lipschitz function with respect to , we therefore have

where g ranges over all the fibers $f( \cdot , y)$ over all . These are all $1$ -bounded and $1$ -Lipschitz with respect to the metric $d(x,y) = (|x-y| \wedge 1).$ It follows from Arzelà-Ascoli and Theorem 1.6 (which gives almost sure weak-* convergence of $\mathscr {D}_{k_2}$ to $\mathscr {D}_{\infty }$ and the almost sure finiteness of $\mathscr {D}_\infty $ ) that

$$\begin{align*}\limsup_{k_2,k_1 \to \infty} \bigg|\int_{\widehat{\Gamma}_{k_7}} f(x,y) \bigl(\mathscr{D}_{k_2}(dx) -\mathscr{D}_{\infty}(dx)\bigr)\widehat{\mathfrak{p}_{k_1}}(dy) \bigg| =0\quad \operatorname{a.s.} \end{align*}$$

Hence, again by Arzelà-Ascoli, taking supremum over all such f, we conclude

$$\begin{align*}\limsup_{k_2,k_1 \to \infty} {d_{\operatorname{BL}}( \mathscr{D}_{k_2} \times \widehat{\mathfrak{p}_{k_1}}, \mathscr{D}_{\infty} \times \widehat{\mathfrak{p}_{k_1}} ) } =0 \quad \operatorname{a.s.} \end{align*}$$

Hence, from Theorem A.2, as $k_1\to \infty $ followed by $k_2 \to \infty ,$ which completes the proof for .

The proof for and is identical.

1.4 Imaginary part of the log-determinant

The following corollary, which handles the imaginary part of the logarithm of the characteristic polynomial, follows from Theorems 1.10 and 1.11 in the same way that Theorem 1.1 followed from them; some adjustments are necessary because $\Im \log X_n(\theta )$ takes values on a (shifted) lattice; see (1.35) below.

Corollary 1.12. There are deterministic constants and almost surely finite random variables so that

The adaptation of the proof needed to handle here the discreteness is somewhat simpler than that needed in the case of lattice valued branching random walks, as described in [Reference Bramson, Ding and ZeitouniBDZ16a, Section 5]; this is because the effects of discreteness here are only felt in the ‘decoration’ process.

The imaginary part of the logarithm of the characteristic polynomial is related to the eigenvalue counting process of the C $\beta $ E. Specifically, we take an increasing version of the map

(1.35) $$ \begin{align} \theta \mapsto n\theta - 2\Im( \log X_n(e^{i\theta}) - \log X_n(1)), \end{align} $$

(cf. Lemma 2.1 below) which in fact counts at all those $\theta $ except $\left \{ -\omega _1,-\omega _2,\dots ,-\omega _n \right \}.$ This has the same law (as a process in $\theta $ ) as

As $\{\log X_n(1)/\sqrt {\log n}\}$ becomes a centered Gaussian (this follows from Proposition 2.2 with some minor adjustments) and

is tight, it follows that

is not tight, and in fact when scaled down by $\sqrt {\log n}$ converges in law to a centered nondegenerate Gaussian.

However, the maximum over all arcs of the centered counting function admits the representation

This motivates understanding the joint convergence of the maximum and the minimum of the imaginary part of the logarithm of the characteristic polynomial, which we do not pursue here.

Note that in the model of Branching Brownian Motion, the analogue of Conjecture 1.13 was recently proved in [Reference Stasiński, Berestycki and MalleinSBM21]; see also [BKL⁺ Reference Chhaibi, Madaule and Najnudel24].

1.5 Related literature

The analysis in this paper falls within the topic of logarithmically correlated fields. Indeed, in the case of $\beta =2$ , it was shown in [Reference Hughes, Keating and O’ConnellHKO01] that the process $ Y_n(z)=\{\log |X_n(z)|\}_{|z|=1}$ is logarithmically correlated in the sense that for any smooth test function on $S^1$ with , the random variable converges in distribution to a centered Gaussian random variable with variance , where is the k-th Fourier coefficient of . See also [Reference Baker and ForresterBF97, Reference Keating and SnaithKS00, Reference WieandWie02] for related results. Note that the above variance expression corresponds to a generalized Gaussian field with correlation having singularity of the form $-\log |\theta -\theta '|$ .

For Gaussian logarithmically correlated fields, the study of the maximum has a long history, going back to the seminal work [Reference BramsonBra78, Reference BramsonBra83] concerning Branching Brownian motion (BBM). Bramson introduced the truncated second moment and barrier methods that are the core tools in all subsequential analysis (see [Reference RobertsRob13] for a modern perspective). Extremal processes for the BBM were constructed in [Reference Aïdékon, Berestycki, Brunet and ShiABBS13] and [Reference Arguin, Bovier and KistlerABK13]. Following some earlier work on tightness and rough structure of the extrema, the extension of the convergence of the maximum to the discrete Gaussian free field in the critical dimension $2$ was obtained in [Reference Bramson, Ding and ZeitouniBDZ16b] and (for the extremal process) in [Reference Biskup and LouidorBL18]. See [Reference BiskupBis20] for an updated account, and [Reference ZeitouniZei16] for an introduction. A convergence result for general log-correlated Gaussian field is presented in [Reference Ding, Roy and ZeitouniDRZ17]; see also [Reference MadauleMad15].

In the physics literature, the notion of freezing in the context of extremal processes of logarithmically correlated fields arose in the seminal work [Reference Fyodorov and BouchaudFB08]. The link between the freezing phenomenon and extremal processes of the decorated, randomly shifted Poisson type was rigorously elucidated in [Reference Subag and ZeitouniSZ15]. The highly influential work [Reference Fyodorov, Hiary and KeatingFHK12b] (see also [Reference Fyodorov, Hiary and KeatingFHK12a]) applied the freezing paradigm to making predictions for the maximum of the logarithm of the modulus of the characteristic polynomial of CUE matrices (see (1.3)) and, using a conjectured dictionary going back to [Reference Keating and SnaithKS00], made predictions concerning the maximum of the Riemann $\zeta $ function over short intervals of the critical axis. This has stimulated much work, both on the random matrix side (which we will review shortly) and on the Riemann side, for which we refer to [Reference BramsonABB⁺19] and [Reference Arguin, Bourgade and RadziwiłłABR23] for the latest progress on verifying the FHK conjectures for the Riemann $\zeta $ function.

On the circular ensembles side, the first verification of the leading order in the FHK prediction was obtained in [Reference Arguin, Belius and BourgadeABB17], followed in short order by the verification of the second-order term [Reference Paquette and ZeitouniPZ18]; both works used explicit computations facilitated by dealing with $\beta =2$ and a decomposition of the log-determinant according to Fourier modes for the former or to spatial approximations for the latter. It was [Reference Chhaibi, Madaule and NajnudelCMN18] who introduced the use of Verblunsky coefficients and the Szegő recursions in (1.5) to not only handle arbitrary $\beta>0$ but also obtain the tightness of $M_n-m_n$ . Along the way, [Reference Chhaibi, Madaule and NajnudelCMN18] derived various Gaussian approximations and barrier estimates that are fundamental for the current paper.

The analysis for C $\beta $ E has natural analogues for Hermitian ensembles of the G $\beta $ E type. We mention, in particular, [Reference Fyodorov and Le DoussalFD16] for an early application of the freezing scenario in that context. More recent work includes [Reference Lambert and PaquetteLP19, Reference Augeri, Butez and ZeitouniABZ23, Reference Claeys, Fahs, Lambert and WebbCFLW21, Reference Bourgade, Mody and PainBMP22, Reference Bourgade, Lopatto and ZeitouniBLZ23]. At this time, the results in the Hermitian setup are much less sharp than for circular ensembles.

An important role in our analysis and results is played by the derivative martingale measure $\mathscr {D}_\infty (d\theta )$ and its total mass $\mathscr {B}_\infty $ , both related to the theory of Gaussian Multiplicative Chaos (GMC); see [Reference Rhodes and VargasRV14] for a review. This is not surprising – the appearance of such martingales in the expression for the law of the maximum of the BBM was discovered already in [Reference Lalley and SellkeLS87]. Subsequentially, it appeared in the analogous studies for Branching Random Walks [Reference AïdékonAïd13], in the context of the Gaussian free field [Reference Duplantier, Rhodes, Sheffield and VargasDRSV14, Reference Biskup and LouidorBL18] and for more general log-correlated fields [Reference MadauleMad15]. As noted above, for a specific log-correlated field on the circle, [Reference RemyRem20] computes the law of the total mass of the associated GMC and confirms the Fyodorov-Bouchaud prediction [Reference Fyodorov and BouchaudFB08] for it.

We also mention that for a related model, [Reference Chhaibi and NajnudelCN19] provide a direct link with the relevant GMC. Their work implies convergence of the random measure when $\gamma =-2$ and $\beta \geq 2$ . After appropriate rescaling, this measure converges to the GMC of [Reference Fyodorov and BouchaudFB08, Reference RemyRem20, Reference Chhaibi and NajnudelCN19] rescaled to be a probability measure. Note that the exponent $\gamma =-2$ never quite appears in the problem we consider, and for any $\beta $ , the relevant $\gamma $ is the positive critical point. When $\beta =2$ , the full Fyodorov-Hiary-Keating conjecture would follow from convergence of $\mathscr {D}_n(\theta )d\theta $ (which up to rescaling) to the same GMC; this is expected to have the same limit as (up to constants).

Very recently, [Reference Lambert and NajnudelLN24] gave a proof of convergence of the random measure to the (subcritical) GMC whose total mass was evaluated in [Reference RemyRem20], for all $\gamma <\sqrt {2\beta }$ . Their techniques might be relevant to the evaluation of the law of $\mathscr {D}_\infty $ , which corresponds to the case of critical GMC.

1.6 A high-level description of the proof

We now provide a high-level description of the proof that glosses over many important details. A detailed description of the proof that includes precise statements is provided in Section 2.

As in [Reference Chhaibi, Madaule and NajnudelCMN18], the key observation is that the recursion (1.10) contains all information needed in order to evaluate the determinant, due to (1.7). For fixed $\theta $ , the variable $\varphi _k(\theta ), k=1,\ldots ,n$ can be well-approximated by a random walk, and further, for all k large, the increments of the random walk are essentially Gaussian.

We explain the strategy toward the proof of Theorem 1.4. We fix large constants $k_1,k_2$ (with $k_1\gg k_2$ ). Instead of directly studying the field , we consider a sublattice $\mathcal {D}_{n/k_1}$ of angles $\theta _j$ of cardinality $\lceil n/k_1\rceil $ , and we associate to each an interval $\widehat {I_{j,n}}$ as in (1.13). We write

and

We claim that the last expression can be approximated as

(1.36) $$ \begin{align} \max_j\Big( \varphi_{k_2}(\theta_j)+ \Delta_{k_2,n/k_1}(\theta_j)+\max_{\theta\in \widehat{I_{j,n}}}\Delta_{n/k_1,n}(\theta)\Big).\end{align} $$

(This is not quite right, and in reality, we will need to consider an intermediate point $n/k_1^+$ with $k_1^+>k_1$ , but we gloss over this detail at this high-level description.)

To analyze the maximum in (1.36), we introduce the field , with and $\Delta $ defined above (1.36), and write (1.36) as

(1.37)

The main contribution to the maximum comes from js with $\Delta _{k_2,n/k_1}(\theta _j)$ large, of the order of $\sqrt {8/\beta }(m_n-\log (k_1k_2))$ . However, the $\Delta _{k_2,n/k_1}(j)$ are far from independent for different j. In order to begin controlling this, we introduce two ‘good events’: a global good event $\mathscr {G}_n$ , which allows us to replace the recursion by one driven by Gaussian variables (called $\mathfrak {z}_t(\theta )$ and taken for convenience in continuous time) and also impose an a priori upper limit on the recursion, and a barrier event $\widehat {\mathscr {R}}$ , which ensures that the Gaussian-driven recursion $\mathfrak {z}_t(\theta )$ stays within a certain entropic envelope. We will also insist that $\mathfrak {z}_{n/k_1}(\theta _j)$ stays within an appropriate window. These steps are similar to what is done in [Reference Chhaibi, Madaule and NajnudelCMN18], and they prepare the ground for the application of the second moment method.

We next claim that the fields $f_{n,j}(\eta )$ converge in distribution to the solution of a system of coupled stochastic differential equations as in (2.52) (again, this is not literally the case and requires some pre-processing in the form of restriction to appropriate events and using $k_1^+$ as before). In particular, the laws of those fields are determined by the Markov kernel $\mathfrak {p}_{k_1}$ . Further and crucially, the fields $f_{n,j}$ can be constructed so that for well-separated js, they are independent. This analysis is contained in Sections 5 and 6.

As in many applications of the second moment method, to allow for some decoupling, it is necessary to condition on $\mathscr {F}_{k_2}$ . We need to find high points of the right side of (1.37). The basic estimate, for a given j, is that with $w_j=\sqrt {8/\beta } \log k_2-\varphi _{k_2}(\theta _j)$ ,

(1.38) $$ \begin{align} \mathbb{P}\Big(\Delta_{k_2,n/k_1}(\theta_j)\sim \sqrt{8/\beta}(m_n-\log (k_1k_2)-v)\mid \mathscr{F}_{k_2}\Big) \sim C \frac{ve^{2v} w_j e^{-2w_j}}{n}.\end{align} $$

This estimate (after some pre-processing) is taken from [Reference Chhaibi, Madaule and NajnudelCMN18]; see Appendix B.

If the variables were an independent family, we would be at this point done, for then we would have that

(1.39)

Hence, we have using independence over different j that

which would then yield Theorem 1.4.

Unfortunately, different js are not independent. We handle that through several Poisson approximations. First, we condition on $\mathscr {F}_{n/k_1}$ and use the ‘two moments suffice’ method of [Reference Arratia, Goldstein and GordonAGG89] to show that the process of near maxima (together with the shape $(\varphi _n(\theta )-\varphi _{n/k_1}(\theta ), \theta \in \widehat {I_{j,n}})$ ) can be well-approximated, as $k_1\to \infty $ , by a Poisson point process of intensity $\mathfrak {m}$ which depends still on $k_1$ ; see (2.59) and Proposition 2.22. In the proof, the independence for well-separated js and the second moment computations play a crucial role. (As mentioned above, we need to replace $k_1$ by $k_1^+\gg k_1$ to make the argument work; for this reason, we introduce a second Poisson approximation; see Section 2.6.)

This is close to the statement of Theorem 1.10, except that the Poisson process we obtained so far has a random intensity, measurable on $\mathscr {F}_{n/k_1^+}$ . Our final step is another standard use of the second moment method to show that this random intensity concentrates; see Section 2.9. As proved earlier in Section 1.3, all our theorems, and in particular Theorem 1.4, follow from Theorem 1.10.

1.7 Glossary

For convenience and easy reference, we record in Tables 1–3 a glossary of notation.

Table 1 Table of large constants. For $p < q$ , $k_p \gg k_q$ .

Table 2 Table of processes, other symbols.

Table 3 Table of events.