Proof. Note that $q=t$ implies $\Psi =\Phi $ and

$$ \begin{align*} \Psi_{\lambda^{(i)},\lambda^{(i+1)}}(A^{(i)},B^{(i+1)};t,t)=\sum_{\nu\in{\mathbb Y}}s_{\lambda^{(i)}/\nu}(A^{(i)})s_{\lambda^{(i+1)}/\nu}(B^{(i+1)}). \end{align*} $$

When $b_i=-$ ,

$$ \begin{align*} s_{\lambda^{(i+1)}/\nu}(0)=\begin{cases}1&\mathrm{if}\ \nu=\lambda^{(i+1)}\\0&\mathrm{otherwise}\end{cases}, \end{align*} $$

and therefore,

$$ \begin{align*} \Psi_{\lambda^{(i)},\lambda^{(i+1)}}(x_i,0;t,t)=s_{\lambda^{(i)}/\lambda^{(i+1)}}(x_i)=\Phi_{\lambda^{(i)},\lambda^{(i+1)}}(x_i,0;t,t). \end{align*} $$

Similarly, when $b_i=+$ ,

$$ \begin{align*} \Psi_{\lambda^{(i)},\lambda^{(i+1)}}(x_i,0;t,t)=s_{\lambda^{(i+1)}/\lambda^{(i)}}(x_i)= \Phi_{\lambda^{(i)},\lambda^{(i+1)}}(x_i,0;t,t).\\[-38pt] \end{align*} $$

Proof. Let $f_{\lambda }(q,t):=(1-t)\sum _{i\geq 1}(q^{\lambda _i}-1)t^{i-1}$ . Then $f_{\lambda }(q,t)=f_{\lambda '}(t,q)$ (see Example 1 in Sect. VI 5 of [Reference Macdonald31]). Also, note that $f(q,t)=\gamma _1(\lambda ;q,\frac {1}{t})-1$ . Finally,

$$ \begin{align*} \gamma_k(\lambda';t,q)=1+f_{\lambda'}\left(t^k,\frac{1}{q^k}\right)=1+f_{\lambda}\left(\frac{1}{q^k},t^k\right)=\gamma_k\left(\lambda;\frac{1}{q},\frac{1}{t}\right). \end{align*} $$

Then the lemma follows.

Proof. By Lemma 4.1, we obtain

$$ \begin{align*} &\left.{\mathbb E}_{\mathrm{Pr}}\left[\prod_{i\in[l+1..r]} \gamma_{l_i}(\lambda^{(i)};q,t)\right]\right|{}_{q=t}\\ &=\left.{\mathbb E}_{\mathrm{Pr}}\left[\prod_{i\in[l+1..r]\cap \mathcal{L}} \gamma_{l_i}(\lambda^{(i)};q,t)\right] \left[\prod_{i\in[l+1..r]\cap \mathcal{R}} \gamma_{l_i}\left(\left[\lambda^{(i)}\right]';\frac{1}{t},\frac{1}{q}\right)\right] \right|{}_{q=t}. \end{align*} $$

Recall that the Macdonald polynomials satisfy (See Page 324 of [Reference Macdonald31])

(4.5) $$ \begin{align} P_{\lambda}(X;q,t)=P_{\lambda}\left(X;\frac{1}{q},\frac{1}{t}\right);\quad Q_{\lambda}(X;q,t)=\left(\frac{t}{q}\right)^{|\lambda|}Q_{\lambda}\left(X;\frac{1}{q},\frac{1}{t}\right). \end{align} $$

We obtain

$$ \begin{align*} &\left.{\mathbb E}_{\mathrm{Pr}}\left[\prod_{i=[l+1..r]\cap \mathcal{L}} \gamma_{l_i}(\lambda^{(i)};q,t)\right] \left[\prod_{i=[l+1..r]\cap \mathcal{R}} \gamma_{l_i}\left(\left[\lambda^{(i)}\right]';\frac{1}{t},\frac{1}{q}\right)\right] \right|{}_{q=t}\\ &=\sum_{\lambda^{(l)},\ldots,\lambda^{(r+1)}\in{\mathbb Y}} \left[\prod_{i=[l+1..r]\cap \mathcal{L}} \gamma_{l_i}(\lambda^{(i)};q,t)\right] \left[\prod_{i=[l+1..r]\cap \mathcal{R}} \gamma_{l_i}\left(\left[\lambda^{(i)}\right]';\frac{1}{t},\frac{1}{q}\right)\right]\\ &\times\left.\mathrm{Pr}(\lambda^{(l)},\ldots,\lambda^{(r+1)}|\lambda^{(l)}=\lambda^{(r+1)}=\emptyset)\right|{}_{q=t}\\&=\frac{1}{\mathcal{Z}}\sum_{\lambda^{(l+1)},\ldots,\lambda^{(r)}\in{\mathbb Y}} \left[\prod_{i=[l+1..r]\cap \mathcal{L}} \gamma_{l_i}(\lambda^{(i)};q,t)\right] \left[\prod_{i=[l+1..r]\cap \mathcal{R}} \gamma_{l_i}\left(\left[\lambda^{(i)}\right]';\frac{1}{t},\frac{1}{q}\right)\right]\\ &\times\left[\prod_{i\in\mathcal{L}}\langle P_{\lambda^{(i)}}(A^{(i)},Y^{(i)};q,t),Q_{\lambda^{(i+1)}}(Y^{(i)},B^{(i+1)};q,t) \rangle_{Y^{(i)}}\right]\\ &\left.\left[\prod_{i\in\mathcal{R}}\langle Q_{[\lambda^{(i)}]'}\left(A^{(i)},Y^{(i)};t,q\right),P_{[\lambda^{(i+1)}]'}\left(Y^{(i)},B^{(i+1)};t,q\right) \rangle_{Y^{(i)}}\right]\right|{}_{q=t,\lambda^{(l)}=\lambda^{(r+1)}=\emptyset}, \end{align*} $$

where for each i, $Y^{(i)}$ is a countable collection of variables, and

$$ \begin{align*} \mathcal{Z}&=\sum_{\lambda^{(l+1)},\ldots,\lambda^{(r)}\in{\mathbb Y}}\left[\prod_{i\in\mathcal{L}}\langle P_{\lambda^{(i)}}(A^{(i)},Y^{(i)};q,t),Q_{\lambda^{(i+1)}}(Y^{(i)},B^{(i+1)};q,t) \rangle_{Y^{(i)}}\right]\\ &\left.\left[\prod_{i\in\mathcal{R}}\langle Q_{[\lambda^{(i)}]'}(A^{(i)},Y^{(i)};t,q),P_{[\lambda^{(i+1)}]'}(Y^{(i)},B^{(i+1)};t,q) \rangle_{Y^{(i)}}\right]\right|{}_{q=t,\lambda^{(l)}=\lambda^{(r+1)}=\emptyset}. \end{align*} $$

For $i\in [l+1..r]$ , let

$$ \begin{align*} \mathbf{E}_i=\begin{cases}{\displaystyle\sum_{\lambda^{(i)}\in {\mathbb Y}}}\gamma_{l_i}(\lambda^{(i)};q,t)P_{\lambda^{(i)}}(A^{(i)},Y^{(i)};q,t)Q_{\lambda^{(i)}}(Y^{(i-1)},B^{(i)};q,t)&\substack{(i-1,i)\\ \in \mathcal{L}\times\mathcal{L}}\\ \displaystyle\sum_{\lambda^{(i)}\in{\mathbb Y}}\gamma_{l_i}([\lambda^{(i)}]';\frac{1}{t},\frac{1}{q}) Q_{\lambda^{(i)}}(Y^{(i-1)},B^{(i)};q,t)Q_{[\lambda^{(i)}]'}(A^{(i)},Y^{(i)};t,q)&\substack{(i-1,i)\\ \in \mathcal{L}\times\mathcal{R}}\\ \displaystyle\sum_{\lambda^{(i)}\in{\mathbb Y}}\gamma_{l_i}(\lambda^{(i)};q,t)P_{[\lambda^{(i)}]'}(Y^{(i-1)},B^{(i)};t,q)P_{\lambda^{(i)}}(A^{(i)},Y^{(i)};q,t)&\substack{(i-1,i)\\ \in \mathcal{R}\times\mathcal{L}}\\ \displaystyle\sum_{\lambda^{(i)}\in{\mathbb Y}}\gamma_{l_i}([\lambda^{(i)}]';\frac{1}{t},\frac{1}{q})P_{[\lambda^{(i)}]'}(Y^{(i-1)},B^{(i)};t,q)Q_{[\lambda^{(i)}]'}(A^{(i)},Y^{(i)};t,q)&\substack{(i-1,i)\\ \in \mathcal{R}\times\mathcal{L}} \end{cases} \end{align*} $$

and

$$ \begin{align*} \mathbf{E}_l&=\begin{cases}P_{\lambda^{(l)}}(A^{(l)},Y^{(l)};q,t) &l\in \mathcal{L}\\ Q_{[\lambda^{(l)}]'}(A^{(l)},Y^{(l)};t,q)& l\in \mathcal{R} \end{cases}\\ \mathbf{E}_{r+1}&=\begin{cases} P_{[\lambda^{(r+1)}]'}(Y^{(r)},B^{(r+1)};t,q)&\ r\in \mathcal{R}\\ Q_{\lambda^{(r+1)}}(Y^{(r)},B^{(r+1)};q,t) & r\in \mathcal{L} \end{cases}. \end{align*} $$

When $\lambda ^{(l)}=\lambda ^{(r+1)}=\emptyset $ and $q=t$ , we have $\mathbf {E}_l=\mathbf {E}_{r+1}=1.$ Then

$$ \begin{align*} &\left.{\mathbb E}_{\mathrm{Pr}} \left[\prod_{i=[l+1..r]\cap \mathcal{L}} \gamma_{l_i}(\lambda^{(i)};q,t)\right] \left[\prod_{i=[l+1..r]\cap \mathcal{R}} \gamma_{l_i}\left(\left[\lambda^{(i)}\right]';t,q\right)\right] \right|{}_{q=t}\\ &=\left.\frac{1}{\mathcal{Z}}\langle \mathbf{E}_{l}\langle \mathbf{E}_{l+1}\ldots\langle \mathbf{E}_{r},\mathbf{E}_{r+1}\rangle_{Y^{((r))}}\ldots \rangle_{Y^{(l+1)}} \rangle_{Y^{(l)}}\right|{}_{q=t,\ \lambda^{(l)}=\lambda^{(r+1)}=\emptyset}. \end{align*} $$

Observe that for $i\in [l+1..r]$ ,

$$ \begin{align*} &\mathbf{E}_i= \begin{cases} D_{-l_i,(A^{(i)},Y^{(i)});q,t}\Pi_{a_{i-1},a_i}((A^{(i)},Y^{(i)}),(Y^{(i-1)},B^{(i)}))&\mathrm{If}\ a_i=L\\ D_{-l_i,(A^{(i)},Y^{(i)});\frac{1}{t},\frac{1}{q}}\Pi_{a_{i-1},a_i}((A^{(i)},Y^{(i)}),(Y^{(i-1)},B^{(i)}))&\mathrm{If}\ a_i=R \end{cases}, \end{align*} $$

where $D_{-l_i,(A^{(i)},Y^{(i)});q,t}$ is the operator defined as in (A.2).

By Proposition A.2, we obtain for $i\in [l+1..r]$ and $q=t$ ,

• • If $a_i=L$ ,

  $$ \begin{align*} \mathbf{E}_i&=\Pi_{a_{i-1},a_i}((A^{(i)},Y^{(i)}),(Y^{(i-1)},B^{(i)}))\\ &\oint\cdots\oint D(W^{(i)};q,t)H(W^{(i)},(A^{(i)},Y^{(i)});q,t)\frac{\Pi_{a_{i-1},a_i}((Y^{(i-1)},B^{(i)}),W^{(i)})}{\Pi_{a_{i-1},a_i}((Y^{(i-1)},B^{(i)}),q^{-1}W^{(i)})}; \end{align*} $$

• • If $a_i=R$ ,

  $$ \begin{align*} \mathbf{E}_i&=\Pi_{a_{i-1},a_i}((A^{(i)},Y^{(i)}),(Y^{(i-1)},B^{(i)}))\\ &\oint\cdots\oint D\left(W^{(i)};\frac{1}{t},\frac{1}{q}\right)H\left(W^{(i)},(A^{(i)},Y^{(i)});\frac{1}{t},\frac{1}{q}\right)\frac{\Pi_{a_{i-1},a_i}((Y^{(i-1)},B^{(i)}),W^{(i)})}{\Pi_{a_{i-1},a_i}((Y^{(i-1)},B^{(i)}),tW^{(i)})}, \end{align*} $$

where each integral contour encloses $0$ and all poles of $H(W^{(i)},(A^{(i)},Y^{(i)});\omega (q,t;a_i))$ ; moreover, if $W^{(i)}=(w^{(i)}_1,w^{(i)}_2,\ldots , w^{(i)}_{l_i})$ , then along the integral contours $|w_j^{(i)}|\leq |\min \{q,t\}w_{j+1}^{(i)}|$ for each $i\in [l_i-1]$ .

Since the integrand in each $\mathbf {E}_i$ is $\Lambda _{Y^{(i)}}$ -projective, by Lemma A.11, we can interchange the order of the residue and Macdonald scalar product and obtain

$$ \begin{align*} &\left.{\mathbb E}_{\mathrm{Pr}}\left[\prod_{i=[l+1..r]} \gamma_{l_i}(\lambda^{(i)};q,t)\right] \right|{}_{q=t} \\&=\frac{1}{\mathcal{Z}}\left(\prod_{i=l+1}^r \Pi_{a_{i-1},a_i}(A^{(i)},B^{(i)})\right)\cdot\oint\langle F_{l}\langle F_{l+1}\ldots\langle F_{r},F_{r+1} \rangle_{Y^{(r)}} \rangle_{Y^{(l+1)}} \rangle_{Y^{(l)}}\\ &\times \left(\prod_{i=[l+1..r]\cap \mathcal{L}}D(W^{(i)};q,t)H(W^{(i)},A^{(i)};q,t)\frac{\Pi_{a_{i-1},a_i}(B^{(i)},W^{(i)})}{\Pi_{a_{i-1},a_i}(B^{(i)},q^{-1}W^{(i)})}\right)\\ &\left.\times\, \left(\prod_{i=[l+1..r]\cap \mathcal{R}}D\left(W^{(i)}; \frac{1}{t},\frac{1}{q}\right)H\left(W^{(i)},A^{(i)};\frac{1}{t},\frac{1}{q}\right)\frac{\Pi_{a_{i-1},a_i}(B^{(i)},W^{(i)})}{\Pi_{a_{i-1},a_i}(B^{(i)},tW^{(i)})}\right)\right|{}_{q=t}. \end{align*} $$

Moreover, for $i\in [l+1..r]$ ,

• • If $a_i=L$ ,

  $$ \begin{align*} F_i&=\Pi_{a_{i-1},a_i}(A^{(i)},Y^{(i-1)})\cdot\Pi_{a_{i-1},a_i}(Y^{(i-1)},Y^{(i)})\cdot \Pi_{a_{i-1},a_i}(Y^{(i)},B^{(i)})\\ &\times H(W^{(i)},Y^{(i)};q,t)\frac{\Pi_{a_{i-1},a_i}(Y^{(i-1)},W^{(i)})}{\Pi_{a_{i-1},a_i}(Y^{(i-1)},q^{-1}W^{(i)})}; \end{align*} $$

• • If $a_i=R$ ,

  $$ \begin{align*} F_i&=\Pi_{a_{i-1},a_i}(A^{(i)},Y^{(i-1)})\cdot\Pi_{a_{i-1},a_i}(Y^{(i-1)},Y^{(i)})\cdot \Pi_{a_{i-1},a_i}(Y^{(i)},B^{(i)})\\ &\times H\left(W^{(i)},Y^{(i)};\frac{1}{t},\frac{1}{q}\right)\frac{\Pi_{a_{i-1},a_i}(Y^{(i-1)},W^{(i)})}{\Pi_{a_{i-1},a_i}(Y^{(i-1)},tW^{(i)})}; \end{align*} $$

and $F_l=F_{r+1}=1.$

By Lemmas A.6 and A.4, we obtain

$$ \begin{align*} \mathcal{F}_r:= \langle F_r,F_{r+1} \rangle_{Y^{(r)}}= \begin{cases} \frac{\Pi_{a_{r-1},a_r}((A^{(r)},W^{(r)}),Y^{(r-1)})}{\Pi_{a_{r-1},a_r}(q^{-1}W^{(r)},Y^{(r-1)})}&\mathrm{If}\ a_r=L\\ \frac{\Pi_{a_{r-1},a_r}((A^{(r)},W^{(r)}),Y^{(r-1)})}{\Pi_{a_{r-1},a_r}(tW^{(r)},Y^{(r-1)})}&\mathrm{If}\ a_r=R \end{cases}. \end{align*} $$

Then the lemma follows by inductively computing the scalar product

$$ \begin{align*} \langle F_{l}\langle F_{l+1}\ldots\langle F_{r},F_{r+1} \rangle_{Y^{(r)}} \rangle_{Y^{(l+1)}} \rangle_{Y^{(l)}} \end{align*} $$

and applying Lemmas 4.4 and 4.5. Note that factors $\Pi _{a_i,a_j}(B^{(i+1)},A^{(j)})$ cancel out with $1/\mathcal {Z}$ since the partition function for the Macdonald process with empty partitions on both ends is

$$ \begin{align*} \prod_{i<j;i,j\in[l,..,r]}\Pi_{a_i,a_j}(B^{(i+1)},A^{(j)}). \\[-49pt]\end{align*} $$

Proof. The Lemma follows from Lemmas A.4 and A.6.

Proof. The lemma follows from Lemma A.6 with

$$ \begin{align*} d_n=\begin{cases} (-1)^{(n+1)(\delta_{a_{i-2},a_{i-1}}-1)}p_n(Y^{(i-2)},B^{(i-1)})+(1-t^{-n})p_n(\frac{q}{W^{(i-1)}}),&a_{i-1}=L\\ (-1)^{(n+1)(\delta_{a_{i-2},a_{i-1}}-1)}p_n(Y^{(i-2)},B^{(i-1)})+(1-q^{n})p_n(\frac{1}{tW^{(i-1)}}),&a_{i-1}=R \end{cases} \end{align*} $$

$$ \begin{align*} u_n&=\begin{cases} (-1)^{(n+1)}\left[p_n(A^{(j)},W^{(j)})-p_n\left(W^{(j)}\xi(a_j)\right)\right]&\mathrm{If}\ a_{i-1}\neq a_j\\ \frac{1-t^n}{1-q^n}\left[p_n(A^{(j)},W^{(j)})-p_n\left(W^{(j)}\xi(a_j)\right)\right]&\mathrm{If}\ a_{i-1}=a_j=L\\ \frac{1-q^{-n}}{1-t^{-n}}\left[p_n(A^{(j)},W^{(j)})-p_n\left(W^{(j)}\xi(a_j)\right)\right]&\mathrm{If}\ a_{i-1}=a_j=R \end{cases}. \\[-40pt]\end{align*} $$

Proof. This proof follows from Lemmas 4.2 and 3.3(2) and $H(X,\{0\};q,t)=1$ and $\Pi _{a,b}(X,\{0\})=\Pi _{a,b}(\{0\},X)=1$ .

Proof. The proof is the same as in [Reference Borodin, Corwin, Gorin and Shakirov5], where Corollary 3.11 is derived from Theorem 3.10. The idea is to apply Lemma 4.2 to an auxiliary Macdonald process $ \left .\mathbb {MP}_{\mathbf {C},\mathbf {D},\mathcal {P}_*,q,t}\right |{}_{\mu ^{(0)}=\mu ^{(r-l+m+1)}=\emptyset }$ , where parameters $\mathbf {C}$ and $\mathbf {D}$ consist of $r-l+m$ variables and the additional indices correspond to copies of $i_j$ for $j\in [m]$ . The expectation formula from Lemma 4.2 restricted to the original process, as in Lemma 4.7, gives the desired formula. For details, see the proof of Corollary 3.11 in [Reference Borodin, Corwin, Gorin and Shakirov5].

Assumption 5.1. Let $\{RYG(l^{(\epsilon )},r^{(\epsilon )}, \underline {a}^{(\epsilon )}, \underline {b}^{(\epsilon )})\}_{\epsilon>0}$ be a sequence of rail yard graphs with the weights of diagonal edges incident with $x=2i$ given with $x_i^{(\epsilon )}$ .

1. 1. Piecewise periodicity of the graph. For a positive integer n and real numbers $V_0<V_1<\ldots <V_m$ , we say that a given sequence of rail yard graphs is n-periodic with transition points $V_0, V_1, \dots ,V_m$ as $\epsilon \rightarrow 0$ if

   1. (a) For each $\epsilon>0$ , there exist integer multiples of n

      $$ \begin{align*} l^{(\epsilon)}=v_0^{(\epsilon)}<v_1^{(\epsilon)}<\ldots<v_m^{(\epsilon)}=r^{(\epsilon)}, \end{align*} $$
      such that $\lim _{\epsilon \rightarrow 0}\epsilon v_p^{(\epsilon )}=V_p,\ \forall p\in \{0\}\cup [m].$

   2. (b) The sequence $ \underline {a}^{(\epsilon )}$ is n-periodic on $\left [l^{(\epsilon )},r^{(\epsilon )}\right ]$ and does not depend on $\epsilon $ . More precisely, there exist $a_1, a_2, \dots , a_n \in \{L,R\}$ such that

      $$ \begin{align*} a_i^{(\epsilon)}=a_{i_{\equiv_n}}, \end{align*} $$
      where $i_{\equiv _n}\in [n]$ is in the same congruence class modulo n as i.

   3. (c) For each $p\in [m]$ , the sequence $ \underline {b}^{(\epsilon )}$ is n-periodic on $(v_{p-1}^{(\epsilon )},v_{p}^{(\epsilon )})$ and does not depend on $\epsilon $ , but it may depend on p. More precisely, there exist $b_{p,1}, b_{p,2}, \dots , b_{p,n} \in \{+,-\}$ such that for $i\in (v_{p-1}^{(\epsilon )},v_{p}^{(\epsilon )})$ ,

      $$ \begin{align*} b_i^{(\epsilon)}=b_{p,i_{\equiv_n}}. \end{align*} $$

2. 2. Periodicity of weights. The weights $x_i^{(\epsilon )}$ are periodic q-volume weights. Precisely, let $k\in [n]$ such that

   $$ \begin{align*} k:=i_{\equiv_n}. \end{align*} $$
   Then $x_i^{(\epsilon )}$ are given with
   $$ \begin{align*} x_i^{(\epsilon)}=\begin{cases} e^{-\epsilon(i-k)}\tau_k,& b_i^{(\epsilon)}=+\\ e^{\epsilon(i-k)}\tau^{-1}_k,& b_i^{(\epsilon)}=- \end{cases}. \end{align*} $$

3. 3. Limit regime.

   1. (a) Assume that $\lim _{\epsilon \rightarrow 0}t=1$ in a such a way that

      $$ \begin{align*} \lim_{\epsilon\rightarrow 0}-\frac{\log t}{n \epsilon}=\beta, \end{align*} $$
      where $\beta $ is a positive real number independent of $\epsilon $ .

   2. (b) Let s be a positive integer and assume that for $d \in [s]$ , the sequence $i_d^{(\epsilon )}\in [l^{(\epsilon )}..r^{(\epsilon )}]$ satisfies

      $$ \begin{align*} \lim_{\epsilon\rightarrow 0}\epsilon i_d^{(\epsilon)}=\chi_d, \end{align*} $$
      for $\chi _1\leq \chi _2\leq \ldots \leq \chi _s$ and that $i_d^{(\epsilon )}\,\mathrm {mod}\, n$ does not depend on $\epsilon $ – that there exist $i^*_1, i^*_2, \dots , i^*_s \in [n]$ such that
      $$ \begin{align*} (i_d^{(\epsilon)})_{\equiv_n} = i^*_d. \end{align*} $$
      If $s=1$ , we drop the index (i.e., we assume that $\lim _{\epsilon \rightarrow 0}\epsilon i^{(\epsilon )}=\chi $ and $(i^{(\epsilon )})_{\equiv _n} = i^*$ ).

Proof. For $p\in [m]$ , $j\in [n]$ and $i\in [l^{(\epsilon )}..r^{(\epsilon )}]$ , let

$$ \begin{align*} I^{(\epsilon)}_{j,p,>i,1}&=\left\{u\in\left[v_{p-1}^{(\epsilon)}+1,v_p^{(\epsilon)}\right]\cap \{n{\mathbb Z}+j\}\cap[i..r^{(\epsilon)}]:b^{(\epsilon)}_u=-,a_u=a_i\right\},\\ I^{(\epsilon)}_{j,p,<i,1}&=\left\{u\in\left[v_{p-1}^{(\epsilon)}+1,v_p^{(\epsilon)}\right]\cap \{n{\mathbb Z}+j\}\cap[l^{(\epsilon)}..i-1]:b^{(\epsilon)}_u=+,a_u=a_i\right\},\\ I^{(\epsilon)}_{j,p,>i,0}&=\left\{u\in\left[v_{p-1}^{(\epsilon)}+1,v_p^{(\epsilon)}\right]\cap \{n{\mathbb Z}+j\}\cap[i..r^{(\epsilon)}]:b^{(\epsilon)}_u=-,a_u\neq a_i\right\},\\ I^{(\epsilon)}_{j,p,<i,0}&=\left\{u\in\left[v_{p-1}^{(\epsilon)}+1,v_p^{(\epsilon)}\right]\cap \{n{\mathbb Z}+j\}\cap[l^{(\epsilon)}..i-1]:b^{(\epsilon)}_u=+,a_u\neq a_i\right\}. \end{align*} $$

Let $N_{j,p,>i,1}^{(\epsilon )}$ , $N_{j,p,<i,1}^{(\epsilon )}$ , $N_{j,p,>i,0}^{(\epsilon )}$ and $N_{j,p,<i,0}^{(\epsilon )}$ be their cardinalities, respectively, and

$$ \begin{align*} &q_{j,p,>i,1}=\max I^{(\epsilon)}_{j,p,>i,1},\qquad q_{j,p,<i,1}=\min I^{(\epsilon)}_{j,p,<i,1},\\ &q_{j,p,>i,0}=\max I^{(\epsilon)}_{j,p,>i,0},\qquad q_{j,p,<i,0}=\min I^{(\epsilon)}_{j,p,<i,0}, \end{align*} $$

where we take the convention that the minimum (resp. maximum) of an empty set is $\infty $ (resp. $-\infty $ ) and define $x_{-\infty }^{(\epsilon )}=x_{\infty }^{(\epsilon )}:=0$ for all $\epsilon \geq 0.$

Then using the above notation, we can rewrite $G^{(\epsilon )}_{1,>i}(W)$ as

$$ \begin{align*} G^{(\epsilon)}_{1,>i}(W)=\prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{c\in I_{j,p,>i,1}^{(\epsilon)}} \left[G_{1,>} (W,e^{-n\epsilon(q_{j,p,>i,1}-c)}x_{q_{j,p,>i,1}}^{(\epsilon)},t)\right], \end{align*} $$

which can be further rewritten as

$$ \begin{align*} G^{(\epsilon)}_{1,>i}(W)=\prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{w_g\in W} \frac{\left(w_g^{-1}x_{q_{j,p,>i,1}}^{(\epsilon)};e^{-n\epsilon}\right)_{N_{j,p,>i,1}^{(\epsilon)}}}{ \left(tw_g^{-1}x_{q_{j,p,>i,1}}^{(\epsilon)};e^{-n\epsilon}\right)_{N_{j,p,>i,1}^{(\epsilon)}} }, \end{align*} $$

where

(5.8) $$ \begin{align} (a;q)_{N}=\prod_{i=0}^{N-1}(1-aq^{i}). \end{align} $$

Similarly, we have

$$ \begin{align*} G^{(\epsilon)}_{1,<i}(W) &=\prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{w_g \in W} \frac{\left( t^{-1}w_gx_{q_{j,p,<i,1}}^{(\epsilon)};e^{-n\epsilon}\right)_{N_{j,p,<i,1}^{(\epsilon)}}}{ \left(w_gx_{q_{j,p,<i,1}}^{(\epsilon)};e^{-n\epsilon}\right)_{N_{j,p,<i,1}^{(\epsilon)}} }, \end{align*} $$

$$ \begin{align*} G^{(\epsilon)}_{0,>i}(W)=\prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{w_g\in W} \frac{\left( -t w_g^{-1}x_{q_{j,p,>i,0}}^{(\epsilon)};e^{-n\epsilon}\right)_{N_{j,p,>i,0}^{(\epsilon)}}}{ \left(-w_g^{-1}x_{q_{j,p,>i,0}}^{(\epsilon)};e^{-n\epsilon}\right)_{N_{j,p,>i,0}^{(\epsilon)}} }, \end{align*} $$

$$ \begin{align*} G^{(\epsilon)}_{0,<i}=\prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{w_g\in W} \frac{\left( -w_gx_{q_{j,p,<i,0}}^{(\epsilon)},e^{-n\epsilon}\right)_{N_{j,p,<i,0}^{(\epsilon)}}}{ \left(-t^{-1}w_gx_{q_{j,p,<i,0}}^{(\epsilon)},e^{-n\epsilon}\right)_{N_{j,p,<i,0}^{(\epsilon)}} }. \end{align*} $$

Assume now that $i=i^{(\epsilon )}$ vary with $\epsilon $ and satisfies the assumptions. For convenience, we continue writing i instead of $i^{(\epsilon )}$ . By Lemma A.12, we obtain as $\epsilon \to 0$ ,

$$ \begin{align*} G^{(\epsilon)}_{1,>i}(W)\sim\prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{w_g\in W} \left(\frac{1-w_g^{-1}x_{q_{j,p,>i,1}}^{(\epsilon)}}{1-e^{-n\epsilon N_{j,p,>i,1}^{(\epsilon)}}w_g^{-1}x_{q_{j,p,>i,1}}^{(\epsilon)}}\right)^{-\frac{\log t}{n\epsilon}}, \end{align*} $$

$$ \begin{align*} G^{(\epsilon)}_{1<i}(W)\sim \prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{w_g\in W} \left(\frac{1-t^{-1}w_gx_{q_{j,p,<i,1}}^{(\epsilon)}}{1-e^{-n\epsilon N_{j,p,<i,1}^{(\epsilon)}}t^{-1}w_gx_{q_{j,p,<i,1}}^{(\epsilon)}}\right)^{-\frac{\log t}{n\epsilon}}, \end{align*} $$

$$ \begin{align*} G^{(\epsilon)}_{0,>i}(W)\sim\prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{w_g\in W} \left(\frac{1+w_g^{-1}x_{q_{j,p,>i,0}}^{(\epsilon)}}{1+e^{-n\epsilon N_{j,p,>i,0}^{(\epsilon)}}w_g^{-1}x_{q_{j,p,>i,0}}^{(\epsilon)}}\right)^{\frac{\log t}{n\epsilon}}, \end{align*} $$

$$ \begin{align*} G^{(\epsilon)}_{0,<i}(W)\sim\prod_{p=1}^{m}\prod_{j=1}^{n}\prod_{w_g\in W} \left(\frac{1+t^{-1}w_gx_{q_{j,p,<i,0}}^{(\epsilon)}}{1+e^{-n\epsilon N_{j,p,<i,0}^{(\epsilon)}}w_gx_{q_{j,p,<i,0}}^{(\epsilon)}}\right)^{\frac{\log t}{n\epsilon}}, \end{align*} $$

where when $W\in {\mathbb C}^k$ , we choose the branch such that when a complex number approaches the positive real line, its argument approaches 0.

Note that for nonempty $I^{(\epsilon )}_{j,p,>i,1}$ , $I^{(\epsilon )}_{j,p,<i,1}$ , $I^{(\epsilon )}_{j,p,>i,0}$ and $I^{(\epsilon )}_{j,p,<i,0}$ , we have

$$ \begin{align*} \lim_{\epsilon\rightarrow 0} n\epsilon N_{j,p,>i,1}^{(\epsilon)} & = V_p-\max\{V_{p-1},\chi\},\\ \lim_{\epsilon\rightarrow 0} n\epsilon N_{j,p,<i,1}^{(\epsilon)}& =\min\{V_p,\chi\}-V_{p-1},\\ \lim_{\epsilon\rightarrow 0} n\epsilon N_{j,p,>i,0}^{(\epsilon)} & =V_p-\max\{V_{p-1},\chi\},\\ \lim_{\epsilon\rightarrow 0} n\epsilon N_{j,p,<i,0}^{(\epsilon)} & =\min\{V_p,\chi\}-V_{p-1}. \end{align*} $$

Also note that

$$ \begin{align*} \lim_{\epsilon\rightarrow 0}x_{q_{j,p,>i,1}}^{(\epsilon)}& =\tau_j^{-1}e^{-\max\{V_{p-1},\chi\}}\mathbf{1}_{\{V_p>\chi,b_{p,j}=-,a_j=a_{i^*}\}},\\ \lim_{\epsilon\rightarrow 0}x_{q_{j,p,<i,1}}^{(\epsilon)}& =\tau_je^{-V_{p-1}}\mathbf{1}_{\{V_{p-1}<\chi,b_{p,j}=+,a_j=a_{i^*}\}},\\ \lim_{\epsilon\rightarrow 0}x_{q_{j,p,>i,0}}^{(\epsilon)}& =\tau_j^{-1}e^{-\max\{V_{p-1},\chi\}}\mathbf{1}_{\{V_{p}>\chi,b_{p,j}=-,a_j \neq a_{i^*}\}},\\ \lim_{\epsilon\rightarrow 0}x_{q_{j,p,<i,0}}^{(\epsilon)}& =\tau_je^{-V_{p-1}}\mathbf{1}_{\{V_{p-1}<\chi,b_{p,j}=+,a_j \neq a_{i^*}\}}. \end{align*} $$

This proves the lemma.

Proof. By Lemma 5.2,

$$ \begin{align*} &\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\left[ \gamma_{k}(\lambda^{(i^{(\epsilon)})};t,t)\right] =\frac{1}{(2\pi\mathbf{i})^{k}}\oint_{\mathcal{C}_1^{(\epsilon)}}\ldots\oint_{\mathcal{C}_k^{(\epsilon)}} \frac{\sum_{i=1}^k\frac{1 }{w_i}}{\left(w_2-w_1\right)\ldots\left(w_k-w_{k-1}\right)}\prod_{i=1}^kdw_i \\ &\prod_{\substack{j\in[(l+1)^{(\epsilon)}..r^{(\epsilon)}],j\geq i^{(\epsilon)}\\b_j=-,a_j=a_{i^*}}}G_{1,>i^{(\epsilon)}}(W,x_j^{(\epsilon)},t) \prod_{\substack{j\in[(l+1)^{(\epsilon)}..r^{(\epsilon)}],j\geq i^{(\epsilon)}\\b_j=-,a_j \neq a_{i^*}}}G_{0,>i^{(\epsilon)}}(W,x_j^{(\epsilon)},t)\\ &\times \prod_{\substack{j\in[l^{(\epsilon)}..r^{(\epsilon)}],j<i^{(\epsilon)}\\b_j=+,a_j= a_{i^*}}}G_{1,<i^{(\epsilon)}}(W,x_j^{(\epsilon)},t) \prod_{\substack{j\in[l^{(\epsilon)}..r^{(\epsilon)}],j<i^{(\epsilon)}\\b_j=+;a_j\neq a_{i^*}}}G_{0,<i^{(\epsilon)}}(W,x_j^{(\epsilon)},t), \end{align*} $$

where for $1\leq i\leq k$ , $\mathcal {C}_i^{(\epsilon )}$ is the integral contour for $w_i$ , and $\mathcal {C}_1^{(\epsilon )},\ldots ,\mathcal {C}_k^{(\epsilon )}$ satisfy the conditions as described in Lemma 4.2. As $\epsilon \rightarrow 0$ , assume that $\mathcal {C}_1^{(\epsilon )},\ldots ,\mathcal {C}_k^{(\epsilon )}$ converge to contours $\mathcal {C}_1,\ldots ,\mathcal {C}_k$ , respectively, such that $\mathcal {C}_1,\ldots \mathcal {C}_k$ are separated from one another and do not cross any of the singularities of the integrand. By Lemma 5.3, we obtain

$$ \begin{align*} &\lim_{\epsilon\rightarrow 0}\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}[ \gamma_{k}(\lambda^{(i^{(\epsilon)})};t,t)]=\frac{1}{(2\pi\mathbf{i})^{k}}\oint_{\mathcal{C}_1}\ldots\oint_{\mathcal{C}_k} \frac{\sum_{i=1}^k\frac{1 }{w_i}}{\left(w_2-w_1\right)\ldots\left(w_k-w_{k-1}\right)}\prod_{i=1}^kdw_i \\ &\times \prod_{w_s\in W}\left[\mathcal{G}_{1,>}\chi(w_x)\mathcal{G}_{1,<\chi}(w_s)\mathcal{G}_{0,>\chi}(w_s)\mathcal{G}_{0,<\chi}(w_s)\right]^{\beta}. \end{align*} $$

Then (5.10) follows from Lemmas A.13 and 5.3.

Proof. Note that $\mathrm {Cov}\left [Q_{k_d}^{(\epsilon )}(\epsilon i_d^{(\epsilon )}) Q_{k_h}^{(\epsilon )}(\epsilon i_h^{(\epsilon )})\right ]$ is equal to

$$ \begin{align*} \frac{\mathbb{E}\left[\gamma_{k_d}(\lambda^{(i_d^{(\epsilon)})};t,t) \gamma_{k_h}(\lambda^{(i_h^{(\epsilon)})};t,t)\right]-\mathbb{E}\left[\gamma_{k_d}(\lambda^{(i_d^{(\epsilon)})};t,t)\right]\mathbb{E}_{\mathrm{Pr}} \left[\gamma_{k_h}(\lambda^{(i_h^{(\epsilon)})};t,t)\right]}{\epsilon^2}. \end{align*} $$

For $W=(w_1,\dots ,w_k)$ , we use abbreviation

$$ \begin{align*} F_i^{(\epsilon)}(W)&=\prod_{\substack{j\in[(l+1)^{(\epsilon)}..r^{(\epsilon)}]\\j\geq i,b_j=-,a_j=a_i}}G_{1,>}(W,x_j^{(\epsilon)},t)\prod_{\substack{j\in[(l+1)^{(\epsilon)}..r^{(\epsilon)}]\\j\geq i,b_j=-,a_j \neq a_i}}G_{0,>}(W,x_j^{(\epsilon)},t)\\ &\times\prod_{\substack{j\in[l^{(\epsilon)}..r^{(\epsilon)}]\\j<i,b_j=+;a_i= a_j}}G_{1,<}(W,x_j^{(\epsilon)},t)\prod_{\substack{j\in[l^{(\epsilon)}..r^{(\epsilon)}]\\j<i,b_j=+;a_i \neq a_j}}G_{0,<}(W,x_j^{(\epsilon)},t)\\ &\times \frac{\sum_{j=1}^{k}\frac{1}{w_j}}{\left(w_2-w_1\right)\ldots\left(w_{k}-w_{k-1}\right)}\prod_{1\leq i<j\leq k }\frac{\left(1-\frac{w_i}{w_j}\right)^2}{\left(1-\frac{w_i}{tw_j}\right)\left(1-\frac{tw_i}{w_j}\right)}. \end{align*} $$

Note that for $\epsilon i^{(\epsilon )}\to \chi $ and $t\to 1$ ,

(5.12) $$ \begin{align} \lim_{\epsilon \to 0}F_{i^{(\epsilon)}}^{(\epsilon)}(W)=\prod_{i=1}^k\mathcal{G}_\chi(w_i) \frac{\sum_{j=1}^{k}\frac{1}{w_j}}{\left(w_2-w_1\right)\ldots\left(w_{k}-w_{k-1}\right)}. \end{align} $$

By Lemma 5.2, we obtain that $\mathrm {Cov}\left [Q_{k_d}^{(\epsilon )}(\epsilon i_d^{(\epsilon )}),Q_{k_h}^{(\epsilon )}(\epsilon i_h^{(\epsilon )})\right ]$ is equal to

$$ \begin{align*} \frac{1}{\epsilon^2(2\pi \mathbf{i})^{k_h+k_d}}\oint_{\mathcal{C}_{1,1}^{(\epsilon)}}\ldots\oint_{\mathcal{C}_{1,k_d}^{(\epsilon)}} \oint_{\mathcal{C}_{2,1}^{(\epsilon)}}\ldots\oint_{\mathcal{C}_{2,k_h}^{(\epsilon)}}& \prod_{\xi \in\{ {d,h}\}}\prod_{i=1}^{k_\xi}dw^{(i_\xi^{(\epsilon)})}_iF^{(\epsilon)}_{i_\xi^{(\epsilon)}}(W^{(i_\xi^{(\epsilon)})})\\ &\times\left[T_{L,L}(W^{(i_d^{(\epsilon)})},W^{(i_h^{(\epsilon)})})-1\right], \end{align*} $$

where $T_{L,L}(Z,W)$ is given by (5.3), $|W^{(i_d^{(\epsilon )})}|=k_d$ , $|W^{(i_h^{(\epsilon )}}|=k_h$ and for $1\leq i\leq k_d$ (resp. $1\leq j\leq k_h$ ), $\mathcal {C}_{1,i}^{(\epsilon )}$ (resp. $\mathcal {C}_{2,j}^{(\epsilon )}$ ) is the integral contour for $w_i^{(i_d^{(\epsilon )})}$ (resp. $w_j^{(i_h^{(\epsilon )})}$ ), and $\mathcal {C}_{1,1}^{(\epsilon )},\ldots ,\mathcal {C}_{1,k_d}^{(\epsilon )}$ , $\mathcal {C}_{2,1}^{(\epsilon )},\ldots ,\mathcal {C}_{2,k_h}^{(\epsilon )}$ satisfy the conditions as described in Lemma 4.2. As $\epsilon \rightarrow 0$ , assume that $\mathcal {C}_1^{(\epsilon )},\ldots ,\mathcal {C}_k^{(\epsilon )}$ converge to contours $\mathcal {C}_1,\ldots ,\mathcal {C}_k$ , respectively, such that $\mathcal {C}_1,\ldots \mathcal {C}_k$ are separated from one another and do not cross any of the singularities of the integrand.

Note that

$$ \begin{align*} T_{L,L}(Z,W)-1=\sum_{\substack{\emptyset\neq S\\S \subset [k_d]\times[k_h]}}\prod_{(i,j)\in S}\frac{(1-t)(t^{-1}-1)z_iw_j}{\left(z_i-t^{-1}w_j \right) \left(z_i-tw_j \right)}. \end{align*} $$

Under Assumption 5.1, we obtain

(5.13) $$ \begin{align} \frac{1}{\epsilon^2}(1-t)(t^{-1}-1)=n^2\beta^2+O(\epsilon). \end{align} $$

Therefore,

$$ \begin{align*} &\frac{1}{\epsilon^2}\left[T_{L,L}(W^{(i_d^{(\epsilon)})},W^{(i_h^{(\epsilon)})})-1\right]\\ &=n^2\beta^2\left[\sum_{(u,v)\in [k_d]\times [k_h] }\frac{w_{u}^{(i_d^{(\epsilon)})} w_{v}^{(i_h^{(\epsilon)})} }{(w_{u}^{(i_d^{(\epsilon)})}-t^{-1} w_{v}^{(i_h^{(\epsilon)})} ) (w_{u}^{(i_d^{(\epsilon)})}-t w_{v}^{(i_h^{(\epsilon)})} )}\right]+o(\epsilon). \end{align*} $$

In the above formula, the main contribution comes from one-element subsets S, whereas the others have a negligible $o(\epsilon )$ contribution. This together with (5.12) gives that the limit of $\mathrm {Cov}\left [Q_{k_d}^{(\epsilon )}(\epsilon i_d^{(\epsilon )}), Q_{k_h}^{(\epsilon )}(\epsilon i_h^{(\epsilon )})\right ]$ is equal to the limit of

$$ \begin{align*} &\frac{n^2\beta^2}{(2\pi\mathbf{i})^{k_d+k_h}}\oint_{\mathcal{C}_{1,1}^{(\epsilon)}}\ldots\oint_{\mathcal{C}_{1,k_d}^{(\epsilon)}} \oint_{\mathcal{C}_{2,1}^{(\epsilon)}}\ldots\oint_{\mathcal{C}_{2,k_h}^{(\epsilon)}} \prod_{\xi \in\{ {d,h}\}}\prod_{i=1}^{k_\xi}dw^{(i_\xi^{(\epsilon)})}_i\\ &\times \prod_{\xi \in \{d,h\}}\left[\prod_{i=1}^{k_{\xi}}\mathcal{G}_{\chi_{\xi}}(w_i^{(i^{(\epsilon)}_\xi)})\right] \frac{\sum_{j=1}^{k_{\xi}}{\Bigl[w_j^{(i^{(\epsilon)}_\xi)}\Bigr]}^{-1}}{(w_2^{(i^{(\epsilon)}_\xi)}-w_1^{(i^{(\epsilon)}_\xi)})\ldots(w_{k_{\xi}}^{(i^{(\epsilon)}_\xi)}-w_{k_{\xi}-1}^{(i^{(\epsilon)}_\xi)})}\\ &\times\left[\sum_{(u,v)\in [k_d]\times [k_h] }\frac{w_{u}^{(i_d^{(\epsilon)})} w_{v}^{(i_h^{(\epsilon)})} }{(w_{u}^{(i_d^{(\epsilon)})}-t^{-1} w_{v}^{(i_h^{(\epsilon)})} ) (w_{u}^{(i_d^{(\epsilon)})}-t w_{v}^{(i_h^{(\epsilon)})} )}+o(\epsilon)\right]. \end{align*} $$

Then by Lemma A.13, we have

$$ \begin{align*} &\lim_{\epsilon\rightarrow 0}\frac{1}{(2\pi\mathbf{i})^{k_h}} \oint_{\mathcal{C}_{2,1}^{(\epsilon)}}\ldots\oint_{\mathcal{C}_{2,k_h}^{(\epsilon)}} \prod_{i=1}^{k_h}dw^{(i_h^{(\epsilon)})}_i\\ &\times \left[\prod_{i=1}^{k_{h}}\mathcal{G}_{\chi_h}(w_i^{(i^{(\epsilon)}_h)})\right] \frac{\sum_{j=1}^{k_h}{\Bigl[w_j^{(i^{(\epsilon)}_h)}\Bigr]}^{-1}}{(w_2^{(i^{(\epsilon)}_h)}-w_1^{(i^{(\epsilon)}_h)})\ldots(w_{k_{h}}^{(i^{(\epsilon)}_h)}-w_{k_{h}-1}^{(i^{(\epsilon)}_h)})}\\ &\times \left[\sum_{(u,v)\in [k_d]\times [k_h] }\frac{w_{u}^{(i_d^{(\epsilon)})} w_{v}^{(i_h^{(\epsilon)})} }{(w_{u}^{(i_d^{(\epsilon)})}-t^{-1} w_{v}^{(i_h^{(\epsilon)})} ) (w_{u}^{(i_d^{(\epsilon)})}-t w_{v}^{(i_h^{(\epsilon)})} )}+o(\epsilon)\right]\\ &=\frac{k_h}{2\pi \mathbf{i}}\oint_{\mathcal{C}_h}\left[ \mathcal{G}_{\chi_h}(w)\ \right]^{k_h\beta}\times\lim_{\epsilon\rightarrow 0}\left[\sum_{u\in [k_d]}\frac{w_{u}^{(i_d^{(\epsilon)})} w }{(w_{u}^{(i_d^{(\epsilon)})}-w ) (w_{u}^{(i_d^{(\epsilon)})}-w )}\right] \frac{dw}{w}. \end{align*} $$

Applying Lemma A.13 again to integrals over $\mathcal {C}_{1,1}^{(\epsilon )}$ , …, $\mathcal {C}_{1,k_d}^{(\epsilon )}$ , we obtain the result.

Proof. Note that

$$ \begin{align*} &\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\left[\prod_{u=1}^{s}Q_{k_{u}}^{(\epsilon)}(\epsilon i_u^{(\epsilon)})\right]\\ &=\frac{1}{\epsilon^s}\sum_{J\subseteq [s]}(-1)^{\left|[s]\setminus J\right|}\left[\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}} \prod_{j\in J}\gamma_{k_j}(\lambda^{(i_j^{(\epsilon)})})\right]\prod_{u\in[s]\setminus J}\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}} \gamma_{k_u}(\lambda^{(i_u^{(\epsilon)})}). \end{align*} $$

When computing $\mathbb {E}\left [\prod _{u=1}^{s}Q_{k_{u}}^{(\epsilon )}(\epsilon i_u^{(\epsilon )})\right ]$ by Lemma 5.2, in the integrand there is a factor

(5.14) $$ \begin{align} \sum_{J\subseteq [s]}(-1)^{\left|[s]\setminus J\right|}\prod_{u<v;u,v\in[s]\setminus J}T_{L,L}(W^{(i_u^{(\epsilon)})},W^{(i_v^{(\epsilon)})}), \end{align} $$

which is by (5.3) equal to

$$ \begin{align*} \sum_{J\subseteq [s]}(-1)^{\left|[s]\setminus J\right|}\prod_{\substack{u,v\in[s]\setminus J\\u<v}}\prod_{\substack{w_j^{(u)}\in W^{(i_u^{(\epsilon)})}\\w_f^{(v)}\in W^{(i_v^{(\epsilon)})}}}\left[1+\frac{(1-t)(t^{-1}-1)w_j^{(u)}w_f^{(v)}}{(w_j^{(u)}-t^{-1}w_f^{(v)}) (w_j^{(u)}-tw_f^{(v)})}\right]. \end{align*} $$

Under the assumptions, (5.13) is true. Let

$$ \begin{align*} \frac{1}{\epsilon^2}\frac{(1-t)(t^{-1}-1)w_j^{(u)}w_f^{(v)}}{\left(w_j^{(u)}-t^{-1}w_f^{(v)}\right) (w_j^{(u)}-tw_f^{(v)})}=C_{u,v,j,f}^{(\epsilon)}, \end{align*} $$

where $C^{(\epsilon )}_{u,v,j,f}$ tends to a constant as $\epsilon \to 0$ . Let

$$ \begin{align*} K_J:=\left\{(u,v,j,f):u<v;u,v\in[s]\setminus J, w_j^{(u)}\in W^{(i_u^{(\epsilon)})},w_f^{(v)}\in W^{(i_v^{(\epsilon)})}\right\}. \end{align*} $$

Then (5.14) is equal to

(5.15) $$ \begin{align} \sum_{J\subseteq [s]}(-1)^{\left|[s]\setminus J\right|}+ \sum_{J\subseteq [s]}(-1)^{\left|[s]\setminus J\right|}\left(\sum_{\emptyset\neq H\subseteq K_J}\epsilon^{2|H|}\prod_{(u,v,j,f)\in H}C_{u,v,j,f}^{(\epsilon)}\right). \end{align} $$

Note that $\sum _{J\subseteq [s]}(-1)^{\left |[s]\setminus J\right |}=0$ . For each fixed $H\subseteq K_{\emptyset }$ , if $H\neq K_{\emptyset }$ , let

$$ \begin{align*} H_0:=\{u\in[s]: \exists v\in[s]\ \mathrm{and } j,f,\ \mathrm{s.t.} (u,v,j,f)\in H,\ \mathrm{or}\ (v,u,j,f)\in H\}. \end{align*} $$

The sum of terms with $\prod _{(u,v,j,f)\in H}C_{u,v,j,f}^{(\epsilon )}$ in (5.15) is

(5.16) $$ \begin{align} \epsilon^{2|H|}\prod_{(u,v,j,f)\in H}C_{u,v,j,f}^{(\epsilon)}\sum_{J\in [s]:J\cap H_0=\emptyset}(-1)^{|[s]\setminus J|}. \end{align} $$

As long as $H_0\neq S$ , the sum of $(-1)^{|[s]\setminus J|}$ over all the subsets of $[s]\setminus H_0$ is 0. Therefore, (5.15) is equal to

$$ \begin{align*} \sum_{\emptyset\neq H\subseteq K_{\emptyset}, H_0=[s]}\epsilon^{2|H|}\prod_{(u,v,j,f)\in H}C_{u,v,j,f}^{(\epsilon)}. \end{align*} $$

1. 1. If s is odd, as $\epsilon \rightarrow 0$ ,

   $$ \begin{align*} \sum_{\emptyset\neq H\subseteq K_{\emptyset}, H_0=[s]}\epsilon^{2|H|}\prod_{(u,v,j,f)\in H}C_{u,v,j,f}^{(\epsilon)}=O(\epsilon^{s+1}); \end{align*} $$
   therefore,
   $$ \begin{align*} \lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon^s}\sum_{J\subseteq [s]}(-1)^{\left|[s]\setminus J\right|}\prod_{u<v;u,v\in[s]\setminus J}T_{L,L}(W^{(i_u^{(\epsilon)})},W^{(i_v^{(\epsilon)})})=0. \end{align*} $$

2. 2. If s is even, as $\epsilon \rightarrow 0$ ,

   $$ \begin{align*} &\sum_{\emptyset\neq H\subseteq K_{\emptyset}, H_0=[s]}\epsilon^{2|H|}\prod_{(u,v,j,f)\in H}C_{u,v,j,f}^{(\epsilon)}\\ &=\epsilon^{s}\sum_{P\in \mathcal{P}_s^2}\prod_{(u,v)\in \mathcal{P}}\prod_{{(j,f):w_j^{(u)}\in W^{(i_u^{(\epsilon)})}, w_j^{(v)}\in W^{(i_v^{(\epsilon)})}}}C_{u,v,j,f}^{(\epsilon)} + O(\epsilon^{s+1})\\ &=\epsilon^{s}\sum_{P\in \mathcal{P}_s^2}\prod_{(u,v)\in \mathcal{P}} \left(\frac{1}{\epsilon^2}T_{L,L}(W^{(i_u^{(\epsilon)})},W^{(i_v^{(\epsilon)})})\right). \end{align*} $$

Then the lemma follows.

Proof. Let $\frac {\kappa }{\epsilon }=y$ . By Theorem 5.4, (2.12) and (4.1), we obtain

$$ \begin{align*} &\lim_{\epsilon\rightarrow 0}\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\int_{-\infty}^{\infty} e^{-n\beta \kappa k}\epsilon h_{M}\left(\frac{\chi}{\epsilon},\frac{\kappa}{\epsilon}\right)d\kappa=\lim_{\epsilon\rightarrow 0}\epsilon^2\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\int_{-\infty}^{\infty} h_{M}\left(\frac{\chi}{\epsilon},y\right)t^{ky}dy \\ &=\lim_{\epsilon\rightarrow 0}\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\frac{2\epsilon^2}{(k\log t)^2}\gamma_{k}(\lambda^{(i^{(\epsilon)})};t,t)=\frac{1}{k^2n^2\beta^2\pi\mathbf{i}}\oint_{\mathcal{C}} \left[\mathcal{G}_{x}(w)\right]^{k\beta}\frac{dw}{w}. \end{align*} $$

To show that the limit, as $\epsilon \rightarrow 0$ , of $\int _{-\infty }^{\infty } e^{-n\beta \kappa k}\epsilon h_{M}\left (\frac {\chi }{\epsilon },\frac {\kappa }{\epsilon }\right )d\kappa $ is non-random, it suffices to show that the limit of its variance is 0. Note that

$$ \begin{align*} &\lim_{\epsilon\rightarrow 0}\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\left[\int_{-\infty}^{\infty} e^{-n\beta \kappa k}\epsilon h_{M}\left(\frac{\chi}{\epsilon},\frac{\kappa}{\epsilon}\right)d\kappa-\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\int_{-\infty}^{\infty} e^{-n\beta \kappa k}\epsilon h_{M}\left(\frac{\chi}{\epsilon},\frac{\kappa}{\epsilon}\right)d\kappa\right]^2\\ &=\lim_{\epsilon\rightarrow 0}\epsilon^4\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\left[\int_{-\infty}^{\infty} e^{-n\beta \kappa k} h_{M}\left(\frac{\chi}{\epsilon},y\right)d y-\mathbb{E}_{\mathrm{Pr}^{(\epsilon)}}\int_{-\infty}^{\infty} e^{-n\beta \kappa k}\epsilon h_{M}\left(\frac{\chi}{\epsilon},y\right)dy\right]^2\\ &=\lim_{\epsilon\rightarrow 0}\frac{\epsilon^6}{(k\log t)^4}\mathrm{Var}\left[Q_k^{(\epsilon)}(\chi)\right]=0, \end{align*} $$

where the last identity follows from Lemma 5.6 and the limit regime stated in Assumption 5.1. Let $\alpha =k\beta $ and consider analytic continuation if necessary. Then the theorem follows.

Assumption 6.2. Let $i,j\in [n]$ and $p_1,p_2 \in [m]$ .

1. 1. Separating condition. For $b_{p_1,i}=-$ , $b_{p_2,j}=+$ and $a_j=a_i$ , it holds that

   (6.11) $$ \begin{align} \tau_{i}^{-1}\tau_{j}& \begin{cases}\geq e^{V_{p_2}-V_{p_1-1}},&{if} \ p_1>p_2\\ >1,&{if} \ p_1=p_2 \end{cases}. \end{align} $$

2. 2. Alternating condition. For $b_{p_1,i}=b_{p_2,j}$ , $a_i=a_j$ , and $\tau _{i}> \tau _{j}$ , it holds that

   $$ \begin{align*} \tau_{i}^{-1}\tau_{j}< e^{V_{p_2-1}-V_{p_1}}. \end{align*} $$

3. 3. Alternatively, we can assume that instead of (6.11), it holds that

   (6.12) $$ \begin{align} \tau_{i}^{-1}\tau_{j}< e^{V_{p_2-1}-V_{p_1}},\quad {if} \ p_1\geq p_2. \end{align} $$

Proof. Let

$$ \begin{align*} \mathcal{G}_{\chi}(w)=C\frac{\prod_{b_i\in \mathcal{Z}_{1,>\chi} \cup\mathcal{Z}_{0,>\chi}\cup\mathcal{Z}_{1,<\chi} \cup\mathcal{Z}_{0,<\chi}}(w-b_i)}{\prod_{a_i\in \mathcal{P}_{1,>\chi} \cup\mathcal{P}_{0,>\chi}\cup\mathcal{P}_{1,<\chi} \cup\mathcal{P}_{0,<\chi}}(w-a_i)}, \end{align*} $$

where $\{a_1<a_2<\cdots <a_k\}$ , $\{b_1<b_2<\cdots <b_k\}$ , and $C\neq 0$ is an absolute constant.

Observe that under Assumptions 6.2 in both cases (i.e., if either (6.11) or (6.12) holds), poles can be divided in three segments: $\{a_1,\dots ,a_{k_1}\}$ , $\{a_{k_1+1},\dots ,a_{k_2}\}$ and $\{a_{k_2+1},\dots ,a_k\}$ so that for each pair of consecutive poles $a_i,a_{i+1}$ , from one of the segments there is a unique zero $b_j\in (a_i,a_{i+1})$ where $j\in [k]$ . For such a pair of poles $a_i,a_{i+1}$ , it is straightforward to check that one of the following two cases occurs:

• • $\lim _{w\rightarrow a_i+}\mathcal {G}_{\chi }(w)=-\infty $ and $\lim _{w\rightarrow a_{i+1}-}\mathcal {G}_{\chi }(w)=+\infty $ ; or

• • $\lim _{w\rightarrow a_i+}\mathcal {G}_{\chi }(w)=+\infty $ and $\lim _{w\rightarrow a_{i+1}-}\mathcal {G}_{\chi }(w)=-\infty $ .

By continuity, $\mathcal {G}_{\chi }(w)$ is a surjection from $(a_i,a_{i+1})$ onto $(-\infty ,\infty )$ . Hence, for each $\zeta \in {\mathbb R}$ , $\mathcal {G}_{\chi }(w)=\zeta ^n$ has at least one root in $(a_i,a_{i+1})$ . Since there are three segments, note that this will give us at least $k-3$ real roots of (6.13).

The following cases might occur:

1. 1. $s^n=C$ . In this case, the equation (6.13) in w has at most $k-1$ roots in the complex plane. But we already have $(k-3)$ real roots; hence, in this case, (6.13) has at most one pair of complex conjugate roots.

2. 2. $s^n\neq C$ . In this case, the equation (6.13) in w has exactly k roots in the complex plane. Again, it is straightforward to check that one of the following two cases occurs:

   • • $\lim _{w\rightarrow a_k+}\mathcal {G}_{\chi }(w)=-\infty $ and $\lim _{w\rightarrow a_{1}-}\mathcal {G}_{\chi }(w)=+\infty $ ; or

   • • $\lim _{w\rightarrow a_k+}\mathcal {G}_{\chi }(w)=+\infty $ and $\lim _{w\rightarrow a_{1}-}\mathcal {G}_{\chi }(w)=-\infty $ .

   By continuity, $\mathcal {G}_{\chi }(w)$ is a surjection from $(-\infty ,a_{1})\cup (a_p,\infty )$ onto $(-\infty ,C)\cup (C,\infty )$ . Hence, for each $s\neq C$ , $\mathcal {G}_{\chi }(w)=s$ has at least one root in $(-\infty ,a_{1})\cup (a_p,\infty )$ . Since (6.13) has at least $(k-2)$ real roots, we deduce that it has at most one pair of complex conjugate roots.

Proof. We only prove part (1) here; part (2) can be proved using exactly the same technique. Assume $\mathcal {R}_{\chi }\neq \emptyset $ and $\mathcal {P}_{1,>\chi }=\emptyset $ ; then $\mathcal {P}_{0,>\chi }\neq \emptyset $ . Let

(6.14) $$ \begin{align} B_1:=\max\mathcal{P}_{0,>\chi}<0. \end{align} $$

• • If $\mathcal {P}_{1,<\chi }\neq \emptyset $ , let

  (6.15) $$ \begin{align} B_2:= \min\mathcal{P}_{1,<\chi}>0. \end{align} $$

• • If $\mathcal {P}_{1,<\chi }= \emptyset $ , let

  (6.16) $$ \begin{align} B_2:=+\infty. \end{align} $$

Then by (5.4)–(5.7) and (5.9), we have

(6.17) $$ \begin{align} \lim_{u\rightarrow B_1+}\mathcal{G}_{\chi}(u)=-\infty. \end{align} $$

Let K be the total number of complex roots (counting multiplicities) in w of $\mathcal {G}_{\chi }(w)=e^{-n\kappa }$ . From the proof of Lemma 6.5, we see that there are at least $K-2$ real roots of $\mathcal {G}_{\chi }(w)=e^{-n\kappa }$ in $(-\infty ,B_1)\cup (B_2,+\infty )$ . If $\mathcal {G}_{\chi }(0)\geq e^{-n\kappa }$ , by (6.17) and the continuity of $\mathcal {G}_{\chi }(w)$ when $w\in (B_1,B_2)$ , we deduce that there is at least one real root of $\mathcal {G}_{\chi }(w)=e^{-n\kappa }$ in $(B_1,0]$ , which contradicts the assumption that $\mathcal {G}_{\chi }(w)=e^{-n\kappa }$ has exactly one pair of nonreal conjugate roots. Then part (1) of the lemma follows.

Proof. We first prove (a) here. By (6.3), (6.6), we obtain

$$ \begin{align*} \frac{\partial \mathcal{H}(\chi,\kappa)}{\partial \kappa} =-\left.\lim_{\epsilon\rightarrow 0+}\frac{1}{\pi}\Im \Theta_{\chi}(\zeta+\mathbf{i}\epsilon)\right|{}_{\zeta=e^{-\kappa}}. \end{align*} $$

By (6.7), (6.9), (6.10), we obtain

(6.20) $$ \begin{align} \frac{\partial \mathcal{H}(\chi,\kappa)}{\partial \kappa} =2\mathbf{1}_{\mathcal{G}_{\chi}(0)>e^{-n\kappa}}-\frac{2}{\pi}\sum_{\xi\in \mathcal{R}_{\chi}}\left[\mathrm{arg}(w_{\xi,\chi}(e^{-\kappa}))-\mathrm{arg}(\xi)\right], \end{align} $$

where the branch of $\mathrm {arg}$ is chosen to have range $(-\pi , \pi ]$ . Hence, we have

$$ \begin{align*} \mathrm{arg}(\xi)=\begin{cases}0&\mathrm{if}\ \xi>0;\\\pi&\mathrm{otherwise}.\end{cases} \end{align*} $$

Under the assumption that $\mathcal {R}_{\chi }\neq \emptyset $ and $\mathcal {G}_{\chi }(0)\neq e^{-n\kappa }$ , the following cases might occur:

1. 1. $\mathcal {P}_{0,>\chi }\neq \emptyset $ . In this case, the number of negative poles in $\mathcal {R}_{\chi }$ is exactly $|\mathcal {P}_{0,>\chi }|$ . From the proof of Lemma 6.5, we see that there are at least $|\mathcal {P}_{0,>\chi }|-1$ negative real roots in $\{w_{\xi ,\chi }(e^{-\kappa })\}_{\xi \in \mathcal {R}_{\chi }}$ . Let $B_1$ be defined as in (6.14).

   1. (a) If $\mathcal {P}_{1,>\chi }\neq \emptyset $ , let

      $$ \begin{align*} B_2:= \min\mathcal{P}_{1,>\chi}>0. \end{align*} $$

   2. (b) If $\mathcal {P}_{1,>\chi }= \emptyset $ and if $\mathcal {P}_{1,<\chi }\neq \emptyset $ , let $B_2$ be defined as in (6.15).

   3. (c) If $\mathcal {P}_{1,>\chi }= \emptyset $ and if $\mathcal {P}_{1,<\chi }= \emptyset $ , let $B_2$ be defined as in (6.16).

   Then by (5.4)–(5.7) and (5.9), we have (6.17). The following cases might occur:

   1. (a) $\mathcal {G}_{\chi }(0)>e^{-n\kappa }$ . Then there exists a unique root in $\{w_{\xi ,\chi }(e^{-\kappa })\}_{\xi \in \mathcal {R}_{\chi }}\cap (B_1,B_2)$ which is negative; in this case, the argument of each negative pole in $\mathcal {R}_{\chi }$ cancels with an argument of a unique negative root in $\{w_{\xi ,\chi }(e^{-\kappa })\}_{\xi \in \mathcal {R}_{\chi }}$ ;

   2. (b) $\mathcal {G}_{\chi }(0)<e^{-n\kappa }$ . Then there exists no root in $\{w_{\xi ,\chi }(e^{-\kappa })\}_{\xi \in \mathcal {R}_{\chi }}\cap (B_1,0)$ ; in this case, there is a unique negative pole in $\mathcal {R}_{\chi }$ whose argument cannot cancel with an argument of a unique negative root in $\{w_{\xi ,\chi }(e^{-\kappa })\}_{\xi \in \mathcal {R}_{\chi }}.$

   In either case, we have that (6.18) holds.

2. 2. $\mathcal {P}_{1,>\chi } \neq \emptyset $ and $\mathcal {P}_{0,>\chi }= \emptyset $ . In this case, there are neither negative poles in $\mathcal {R}_{\chi }$ nor negative real roots in $\{w_{\xi ,\chi }(e^{-\kappa })\}_{\xi \in \mathcal {R}_{\chi }}$ . By Lemma 6.6, we have $\mathcal {G}_{\chi }(0)>0$ . Then (6.18) follows from (6.20).

Now we prove (b). Following the same argument as the proof of part(a), we obtain

$$ \begin{align*} \frac{\partial \mathcal{H}(\chi,\kappa)}{\partial \kappa} = 2\left[\mathbf{1}_{\mathcal{G}_{\chi}(0)> e^{-n\kappa}}+\mathbf{1}_{\mathcal{G}_{\chi}(\infty)> e^{-n\kappa}}-\frac{\mathrm{arg}(\mathbf{w})}{\pi}\right], \end{align*} $$

where $\mathbf {w}$ is some non-real root of $\mathcal {G}_{\chi }(w)=e^{-n\kappa }$ . Depending on the value of $\mathbf {1}_{\mathcal {G}_{\chi }(0)> e^{-n\kappa }}+\mathbf {1}_{\mathcal {G}_{\chi }(\infty )> e^{-n\kappa }}$ , $\mathbf {w}$ may be chosen to be the root in the upper half plane or the lower half plane with appropriate branch of $arg(\cdot )$ such that $\frac {\partial \mathcal {H}}{\partial \kappa }\in [0,2]$ , as in (2.9). Then (6.19) follows.