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Fluctuations of \(\beta \)-Jacobi product processes

Abstract

We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index \(\beta = 1,2,4\) respectively) where time corresponds to the number of terms in the product. More generally, we consider the \(\beta \)-Jacobi product process obtained by extrapolating to arbitrary \(\beta > 0\). For fixed time (i.e. number of factors is constant), we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When \(\beta = 2\), our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann–Burda–Kieburg and Liu–Wang–Wang across time. In the arbitrary \(\beta > 0\) case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural \(\beta \)-generalization of the interpolating process.

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Acknowledgements

I would like to thank my advisor Vadim Gorin for suggesting the study of matrix products in orthogonal and symplectic symmetry classes, for useful discussions, and giving feedback on several drafts. I also thank the anonymous referees for many helpful comments which improved this text. The author was partially supported by National Science Foundation Grant DMS-1664619.

Funding

The author was partially supported by NSF Grant DMS-1664619 (see acknowledgments).

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Appendices

Appendix A

We demonstrate that \(\beta \)-Jacobi product processes with certain parameters can be realized as the squared singular values of products of truncated Haar-distributed \({\mathbb {U}}_\beta \) matrices for \(\beta = 1,2,4\).

Fix \(\theta \in \{1/2,1,2\}\), and positive integer parameters \(L_1,L_2,\ldots \), N, \(N_1,N_2,\ldots ,\) such that

$$\begin{aligned} N \le N_T, \quad \max (N_{T-1},N_T) \le L_T, \quad \quad T \in {\mathbb {Z}}_+ \end{aligned}$$

where \(N_0 := N\). Let \(U_1,U_2,\ldots \) be independent random matrices such that \(U_T\) is a random Haar \({\mathbb {U}}_{2\theta }(L_T)\) matrix. Let \(X_T\) be an \(N_T \times N_{T-1}\) submatrix (or truncation) of \(U_T\),

$$\begin{aligned} Y_T := X_T \cdots X_1, \end{aligned}$$

and \(\mathbf {y}^{(T)} \in {\mathcal {R}}^N\) be the ordered vector of eigenvalues of \(Y_T^*Y_T\).

Theorem A.1

Suppose \(\mathbf {x}^{(1)},\mathbf {x}^{(2)},\ldots \) are independent random vectors in \({\mathcal {R}}^N\) such that \(\mathbf {x}^{(T)} \sim \P _\theta ^{\alpha _T,M_T,N}\) where \(\alpha _T = N_T - N + 1\) and \(M_T = L_T - N_T\). Then \(\left( \mathbf {y}^{(T)}\right) _{T\in {\mathbb {Z}}^+}\) is equal in distribution to

$$\begin{aligned} \left( \mathbf {x}^{(1)} \boxtimes _{2\theta } \cdots \boxtimes _{2\theta } \mathbf {x}^{(T)}\right) _{T \in {\mathbb {Z}}^+}. \end{aligned}$$

Lemma A.2

Let \(n,n_1,n_2,m \in {\mathbb {Z}}_+\) such that \(n \le n_i \le m\), \(i = 1,2\). Suppose X and \(\widetilde{X}\) are respectively \(n_2 \times n_1\) and \(n_2\times n\) truncations of a random Haar \({\mathbb {U}}_{2\theta }(m)\) matrix. Let W be a fixed \(n_1 \times n\) matrix and \(\widetilde{W} := (W^*W)^{1/2}\). If \(\sigma (A) \in {\mathcal {R}}^N\) denotes the singular values of a matrix A, then \(\sigma (XW) \overset{d}{=} \sigma (\widetilde{X} \widetilde{W})\).

Proof

Let \(P_{a\times b}\) denote the \(a\times b\) matrix with the \(\min (a,b)\times \min (a,b)\) identity matrix in the upper left corner and 0 elsewhere. The singular value decomposition of M gives

$$\begin{aligned} W = U P_{n_1\times n} \Sigma V^* \end{aligned}$$

where \(\Sigma = \mathrm {diag}(\sigma (W))\), \(U \in {\mathbb {U}}_{2\theta }(n_1),V \in {\mathbb {U}}_{2\theta }(n)\). Then

$$\begin{aligned} (XW)^*(XW) = V \Sigma P_{n\times n_1} U^* X^* X U P_{n_1\times n} \Sigma V^*. \end{aligned}$$

Observe that

$$\begin{aligned} P_{n\times n_1} U^* X^* X U P_{n_1\times n} \overset{d}{=} \widetilde{X}^* \widetilde{X} \overset{d}{=} V^* \widetilde{X}^* \widetilde{X} V \end{aligned}$$

using the fact that the distributions of X and \(\widetilde{X}\) are invariant under right translation by \({\mathbb {U}}_{2\theta }\). Thus

$$\begin{aligned} (XW)^*(XW) \overset{d}{=} V \Sigma V^* \widetilde{X}^* \widetilde{X} V \Sigma V^* = (\widetilde{X} \widetilde{W})^* \widetilde{X} \widetilde{W} \end{aligned}$$

which proves the lemma. \(\square \)

Proof of Theorem A.1 Set

$$\begin{aligned} \widetilde{\mathbf {y}}^{(T)} := \mathbf {x}^{(1)} \boxtimes _{2\theta } \cdots \boxtimes _{2\theta } \mathbf {x}^{(T)}. \end{aligned}$$

Let \(\widetilde{X}_1,\widetilde{X}_2,\ldots \) be independent such that \(\widetilde{X}_T\) is an \(N_T \times N\) truncation of a Haar \({\mathbb {U}}_{2\theta }(M_T)\) matrix, and

$$\begin{aligned} \widetilde{Y}_T := (\widetilde{X}_T^*\widetilde{X}_T)^{1/2} \cdots (\widetilde{X}_1^*\widetilde{X}_1)^{1/2}. \end{aligned}$$

By Proposition 3.3 and the definition of \(\boxtimes _{2\theta }\), \(\widetilde{\mathbf {y}}^{(T)}\) is the squared singular values vector of \(\widetilde{Y}_T\). We have \(\mathbf {y}^{(1)} \overset{d}{=} \widetilde{\mathbf {y}}^{(1)}\), and for any fixed \(\mathbf {u} \in {\mathcal {U}}^N\),

$$\begin{aligned} (\mathbf {y}^{(T)}|\mathbf {y}^{(T-1)} = \mathbf {u}) \overset{d}{=} (\widetilde{\mathbf {y}}^{(T)}|\widetilde{\mathbf {y}}^{(T-1)} = \mathbf {u}) \end{aligned}$$

by Lemma A.2, thus completing our proof. \(\square \)

Appendix B. Observables of Schur Processes

We derive a contour integral formula for joint moments of a special case of Schur processes—Macdonald processes in the case \(q = t\).

For \(q = t\), the Macdonald symmetric functions become the Schur functions

$$\begin{aligned} s_\lambda (X) := P_\lambda (X;t,t) = Q_\lambda (X;t,t), \quad s_{\lambda /\mu }(X) := P_{\lambda /\mu }(X;t,t) = Q_{\lambda /\mu }(X;t,t) \end{aligned}$$

which are independent of t. Thus properties for Macdonald symmetric functions are inherited by the Schur functions. For example, for any countable set of variables XY, (2.2) implies

$$\begin{aligned} s_{\lambda /\nu }(X,Y) = \sum _{\mu \in {\mathbb {Y}}} s_{\lambda /\mu }(X) s_{\mu /\nu }(Y). \end{aligned}$$
(B.1)

The Schur functions have an explicit form

$$\begin{aligned} s_\lambda (x_1,\ldots ,x_n) = \frac{\det \begin{pmatrix} x_i^{\lambda _j + n - j} \end{pmatrix}_{i,j=1}^n}{\prod _{1 \le i < j \le n} (x_i - x_j) }, \end{aligned}$$
(B.2)

for details see [42, Chapter I, Sections 3  & 4]

Definition B.1

Suppose \(\mathbf {a} := (a_1,\ldots ,a_N) \in {\mathbb {R}}_+^N\), \(\mathbf {b} \in (b_1,\ldots ,b_M) \in {\mathbb {R}}_+^M\) such that \(a_ib_j < 1\) for \(1 \le i \le N\), \(1 \le j \le M\). Let \(\mathbb {SP}_{\mathbf {a},\mathbf {b}}\) denote the measure on \({\mathbb {Y}}^M = {\mathbb {Y}}\times \cdots \times {\mathbb {Y}}\) where

$$\begin{aligned} {\mathbb {SP}}_{\mathbf {a},\mathbf {b}}(\mathbf {\lambda }) := \prod _{\begin{array}{c} 1 \le i \le N \\ 1 \le j \le M \end{array}} (1 - a_i b_j) \cdot s_{\lambda ^M}(a_1,\ldots ,a_N) s_{\lambda ^M/\lambda ^{M-1}}(b_M) \cdots s_{\lambda ^2/\lambda ^1}(b_2) s_{\lambda ^1}(b_1) \end{aligned}$$
(B.3)

for \(\mathbf {\lambda } := (\lambda ^1,\ldots ,\lambda ^M) \in {\mathbb {Y}}^M\).

Remark 9

The measure \(\mathbb {SP}_{\mathbf {a},\mathbf {b}}\) is the \(q = t\) case of the so-called ascending Macdonald process. We have

$$\begin{aligned} \sum _{\lambda ^1,\ldots ,\lambda ^M \in {\mathbb {Y}}} s_{\lambda ^M}(a_1,\ldots ,a_N) s_{\lambda ^M/\lambda ^{M-1}}(b_M) \cdots s_{\lambda ^2/\lambda ^1}(b_2) s_{\lambda ^1}(b_1) = \prod _{\begin{array}{c} 1 \le i \le N \\ 1 \le j \le M \end{array}} \frac{1}{1 - a_i b_j} \end{aligned}$$

as a consequence of (B.1) and the \(q = t\) case of (2.1).

We prove the following contour integral formula for joint expectations of \({\varvec{p}}_t\) (recall Definition 4.4).

Theorem B.2

Suppose \(\mathbf {a} := (a_1,\ldots ,a_N) \in {\mathbb {R}}_+^N\), \(\mathbf {b} \in (b_1,\ldots ,b_M) \in {\mathbb {R}}_+^M\) such that \(a_ib_j < 1\) for \(1 \le i \le M\), \(1 \le j \le N\). For real \(t_1,\ldots ,t_m > 0\), integers \(1 \le n_m \le \cdots \le n_1 \le M\), and \(\mathbf {\lambda } \sim \mathbb {SP}_{\mathbf {a},\mathbf {b}}\), we have

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m {\varvec{p}}_{t_i}(\lambda ^{n_i})\right]&= \frac{1}{(2\pi \mathbf {i})^m} \oint \cdots \oint \prod _{1 \le i < j \le m} \frac{(z_j - z_i)(t_i z_j - t_j z_i)}{(t_i z_j - z_i)(z_j - t_j z_i)} \prod _{i=1}^m \\&\quad \left( \prod _{\ell =1}^N \frac{z_i - a_\ell }{z_i - t_i a_\ell } \cdot \prod _{\ell =1}^{n_i} \frac{1 - t_i^{-1} b_\ell z_i}{1 - b_\ell z_i} \right) \frac{dz_i}{z_i} \end{aligned}$$

where the \(z_i\)-contour \({\mathfrak {Y}}_i\) is positively oriented around \(0, \{t_ia_\ell \}_{\ell =1}^N\) but does not encircle \(\{b_\ell ^{-1}\}_{\ell =n_i}^M\), and is encircled by \(t_j^{-1} {\mathfrak {Y}}_j\), \(t_i {\mathfrak {Y}}_j\) for \(j > i\); given that such contours exist.

Proof

Let \(D_t^{x_1,\ldots ,x_n}\) act on functions in \((x_1,\ldots ,x_n)\) by

$$\begin{aligned} D_t^{x_1,\ldots ,x_n} := (1 - t^{-1})\sum _{i=1}^n \prod _{j \ne i} \frac{x_i - t^{-1}x_j}{x_i - x_j} T_{t,x_i} + t^{-n} \end{aligned}$$

where \(T_{t,x_i}\) is the t-shift operator

$$\begin{aligned} T_{t,x_i}: f(x_1,\ldots ,x_n) \mapsto f(x_1,\ldots ,x_{i-1},tx_i,x_{i+1},\ldots ,x_n). \end{aligned}$$

Using (B.2), we have

$$\begin{aligned} D_t^{x_1,\ldots ,x_n} s_\lambda (x_1,\ldots ,x_n) = {\varvec{p}}_t(\lambda ) s_\lambda (x_1,\ldots ,x_n), \quad \quad \lambda \in {\mathbb {Y}}_n. \end{aligned}$$

Then (B.1) and (B.3) imply

$$\begin{aligned} \prod _{\begin{array}{c} 1 \le i \le N \\ 1 \le j \le M \end{array}} (1 - a_i b_j) \cdot D_{t_m}^{b_1,\ldots ,b_{n_m}} \cdots D_{t_1}^{b_1,\ldots ,b_{n_1}} \prod _{\begin{array}{c} 1 \le i \le N \\ 1 \le j \le M \end{array}} \frac{1}{1 - a_i b_j} = {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m {\varvec{p}}_{t_i}(\lambda ^{n_i}) \right] . \end{aligned}$$

On the other hand, if \(n \le M\) and f is analytic in a neighborhood of \(\{0\} \cup \{b_i\}_{i=1}^n\) such that f(tw)/f(w) is also analytic in this neighborhood, then by residue expansion,

$$\begin{aligned} D_t^{b_1,\ldots ,b_n} \prod _{i=1}^M f(b_i) = \prod _{i=1}^M f(b_i) \cdot \frac{1}{2\pi \mathbf {i}} \oint \left( \prod _{j=1}^n \frac{w - t^{-1} b_j}{w - b_j} \right) \frac{f(tw)}{f(w)} \frac{dw}{w} \end{aligned}$$

where the contour is positively oriented around 0, \(\{b_i\}_{i=1}^n\), but no other poles of the integrand. By iterating, we obtain

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m {\varvec{p}}_{t_i}(\lambda ^{n_i}) \right]&= \prod _{\begin{array}{c} 1 \le i \le N \\ 1 \le j \le M \end{array}} (1 - a_i b_j) \cdot D_{t_m}^{b_1,\ldots ,b_{n_m}} \cdots D_{t_1}^{b_1,\ldots ,b_{n_1}} \prod _{\begin{array}{c} 1 \le i \le N \\ 1 \le j \le M \end{array}} \frac{1}{1 - a_i b_j} \\&= \frac{1}{(2\pi \mathbf {i})^m} \oint \cdots \oint \prod _{1 \le i < j \le m} \frac{(w_i - w_j)(w_i - \frac{t_j}{t_i}w_j)}{(w_i - \frac{1}{t_i} w_j)(w_i - t_j w_j)} \\&\quad \prod _{i=1}^m \left( \prod _{j=1}^{n_i} \frac{w_i - \frac{1}{t_i} b_j}{w_i - b_j} \prod _{j=1}^N \frac{1 - a_j w_i}{1 - t_i a_j w_i} \right) \frac{dw_i}{w_i} \end{aligned}$$

where the \(w_j\)-contour is positively oriented around 0, \(\{b_j\}_{j=1}^{n_i}\), but no other poles of the integrand, and enclosed by \(t_i w_i\), \(t_j^{-1} w_i\) for \(i < j\); assuming such contours exist. Taking \(z_i = w_i^{-1}\) completes the proof. \(\square \)

Appendix C

We recall the notion of cumulants and some basic properties.

Definition C.1

For any finite set S, let \(\Theta _S\) be the collection of all set partitions of S, that is

$$\begin{aligned} \Theta _S := \left\{ \{S_1,\ldots ,S_d\}: d > 0, ~\bigcup _{i=1}^d S_i = S, ~S_i \cap S_j = \emptyset ~\forall i \ne j,~S_i \ne \emptyset ~\forall i \in [[1,d]] \right\} . \end{aligned}$$

For a random vector \(\mathbf {u} = (u_1,\ldots ,u_m)\) and any \(v_1,\ldots ,v_\nu \in \{u_1,\ldots ,u_m\}\), define the (order \(\nu \)) cumulant

$$\begin{aligned} \kappa (v_1,\ldots ,v_\nu ) := \sum _{\begin{array}{c} d > 0 \\ \{S_1,\ldots ,S_d\} \in \Theta _{[[1,\nu ]]} \end{array}} (-1)^{d-1} (d-1)! \prod _{\ell =1}^d {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i \in S_\ell } v_i \right] . \end{aligned}$$
(C.1)

The definition implies that for any random vector \(\mathbf {u}\), the existence of all cumulants of order up to \(\nu \) is equivalent to the existence of all moments of order up to \(\nu \). Note that the cumulants of order 2 are exactly the covariances:

$$\begin{aligned} \kappa (v_1,v_2) = \mathrm {Cov}(v_1,v_2). \end{aligned}$$

We can also express the cumulant as

$$\begin{aligned} \kappa (v_1,\ldots ,v_\nu ) = (-\mathbf {i})^\nu \left. \frac{\partial ^\nu }{\partial t_1 \cdots \partial t_m} \log {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \exp \left( \mathbf {i}\sum _{j=1}^\nu t_j v_j \right) \right] \right| _{t_1 = \cdots = t_\nu = 0}. \end{aligned}$$
(C.2)

For further details see [48, Sections 3.1  & 3.2] wherein the agreement between (C.1) and (C.2) is shown by taking (C.2) as the definition and proving (C.1). Note that (C.2) implies

Lemma C.2

A random vector is Gaussian if and only if all cumulants of order \(\ge 3\) vanish.

We have a useful inversion lemma

Lemma C.3

If K and E are complex-valued functions on the set of nonempty subsets of \([[1,\nu ]]\) such that

$$\begin{aligned} E(S) = \sum _{\begin{array}{c} d > 0 \\ \{S_1,\ldots ,S_d\} \in \Theta _S \end{array}} \prod _{i=1}^d K(S_i) \quad \quad \text{ for } \text{ all } \text{ nonempty } S \subset [[1,\nu ]]\text{, } \end{aligned}$$

then

$$\begin{aligned} K(S) = \sum _{\begin{array}{c} d > 0 \\ \{S_1,\ldots ,S_d\} \in \Theta _S \end{array}} (-1)^{d-1} (d-1)! \prod _{i=1}^d E(S_i). \end{aligned}$$
(C.3)

Proof

Let \(E_{n_1,\ldots ,n_\nu } \in {\mathbb {C}}\) with \(E_{0,\ldots ,0} = 1\). Define the formal power series

$$\begin{aligned} E(t_1,\ldots ,t_\nu )&:= \sum _{n_1,\ldots ,n_\nu \ge 0} \frac{E_{n_1,\ldots ,n_\nu }}{n_1! \cdots n_\nu !} t_1^{n_1} \cdots t_\nu ^{n_\nu } \\ K(t_1,\ldots ,t_\nu )&:= \sum _{n_1,\ldots ,n_\nu \ge 0} \frac{K_{n_1,\ldots ,n_\nu }}{n_1! \cdots n_\nu !} t_1^{n_1} \cdots t_\nu ^{n_\nu } := \log E(t_1,\ldots ,t_\nu ). \end{aligned}$$

For each nonempty \(S \subset [[1,\nu ]]\), let \(K_{n_1,\ldots ,n_\nu } = K(S)\) if \(n_j = \mathbf {1}_{j\in S}\). The relation \(E(\mathbf {t}) = e^{K(\mathbf {t})}\) implies

$$\begin{aligned} E_{n_1,\ldots ,n_\nu } = \sum _{\begin{array}{c} d > 0 \\ \{S_1,\ldots ,S_d\} \in \Theta _S \end{array}} \prod _{\ell =1}^d K(S_\ell ) = E(S) \end{aligned}$$

for any nonempty \(S \subset [[1,\nu ]]\) where \(n_j = \mathbf {1}_{j\in S}\). The relation \(K(\mathbf {t}) = \log E(\mathbf {t})\) now implies (C.3). \(\square \)

Appendix D. Convergence of Laplace transform

We provide calculations which demonstrate that the Laplace transform of the interpolating process converges to that of the picket fence, a Gaussian random variable, and the Airy point process in the respect regimes discussed in Sect. 1.2. We consider each regime in a separate sub-appendix below. The key input for each of these convergence results is the (fixed-time) formula for the Laplace transform of the interpolating process from Theorem 1.4: For \(c_1,\ldots ,c_m > 0\), we have

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{c_i {\mathfrak {x}}_j^{(\widehat{T})}} \right]&= \frac{1}{(2\pi \mathbf {i})^m} \oint \cdots \oint \det \left[ \frac{1}{u_i - u_j - c_j} \right] _{i,j=1}^m \prod _{i=1}^m\nonumber \\&\quad \frac{\Gamma (-u_i-c_i)}{\Gamma (-u_i)} \exp \left[ -\widehat{T} \left( \frac{c_i(c_i + 1)}{2} + c_i u_i \right) \right] du_i \end{aligned}$$
(D.1)

where the \(u_i\)-contour is a positively oriented contour around \(-c_i,-c_i+1,-c_i+2,\ldots \) which starts and ends at \(+\infty \), and the \(u_j\) contour contains \(u_i + c_i\) and \(u_i - c_j\) for \(j > i\).

1.1 D.1 \(\widehat{T} \rightarrow \infty \): Picket Fence Statistics

We demonstrate the convergence of \(\{\widehat{T}^{-1} {\mathfrak {x}}_i^{(\widehat{T})}\}_{i=1}^\infty \) to picket fence statistics \(\{-i+\tfrac{1}{2}\}_{i=1}^\infty \) by way of Laplace transforms. More precisely, we show the convergence

$$\begin{aligned} \sum _{i=1}^\infty \delta _{\widehat{T}^{-1} {\mathfrak {x}}_i^{(\widehat{T})}} \rightarrow \sum _{i=1}^\infty \delta _{-i+\frac{1}{2}} \end{aligned}$$
(D.2)

weakly in probability, as \(\widehat{T} \rightarrow \infty \). This convergence is a consequence of the following:

Theorem D.1

For \(0< \widehat{c} < 1\)

$$\begin{aligned} \lim _{\widehat{T}\rightarrow \infty } {{\,\mathrm{{\mathbb {E}}}\,}}\left( \sum _{j=1}^\infty e^{\widehat{c}\widehat{T}^{-1} {\mathfrak {x}}_j} \right) = \sum _{j=1}^\infty e^{\widehat{c}(-j + \frac{1}{2})} \end{aligned}$$
(D.3)

and

$$\begin{aligned} \lim _{\widehat{T}\rightarrow \infty } \mathrm {Var}\left( \sum _{j=1}^\infty e^{\widehat{c}\widehat{T}^{-1} {\mathfrak {x}}_j} \right) = 0. \end{aligned}$$
(D.4)

Indeed, Theorem D.1 implies the intermediary weak convergence in probability

$$\begin{aligned} \sum _{i=1}^\infty e^{\varepsilon \widehat{T}^{-1} {\mathfrak {x}}_i^{(\widehat{T})}} \delta _{\widehat{T}^{-1} {\mathfrak {x}}_i^{(\widehat{T})}} \rightarrow \sum _{i=1}^\infty e^{\varepsilon (-i+\frac{1}{2})} \delta _{-i+\frac{1}{2}} \end{aligned}$$

of finite measures as \(\widehat{T} \rightarrow \infty \) for some \(\varepsilon > 0\) small and fixed, see e.g. [8, Lemma 2.1.7]. This clearly implies (D.2).

Proof

Recall that

$$\begin{aligned}{\mathrm {Res}}_{z=-n} \Gamma (z) = \frac{(-1)^n}{n!}, \quad \quad n = 0,1,2,\ldots . \end{aligned}$$

Starting from (D.1) with \(m = 1\) and evaluating residues, we have

$$\begin{aligned} \begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \sum _{j=1}^\infty e^{c{\mathfrak {x}}_j^{(\widehat{T})}} \right]&= - \frac{1}{2\pi \mathbf {i}} \oint \frac{\Gamma (-u-c)}{\Gamma (-u)} \exp \left[ -\widehat{T} \left( \frac{c(c+ 1)}{2} + cu \right) \right] \frac{du}{c} \\&= \sum _{n=0}^\infty \frac{(-1)^n}{n!} \frac{1}{c \Gamma (c-n)} \exp \left[ -\widehat{T} cn + \widehat{T} \frac{c^2}{2} - \widehat{T} \frac{c}{2} \right] \\&= \sum _{n=0}^\infty \frac{(-1)^n \sin (\pi (c-n))}{\pi c} \frac{\Gamma (n+1 - c)}{\Gamma (n+1)} \exp \left[ -\widehat{T} cn + \widehat{T} \frac{c^2}{2} - \widehat{T} \frac{c}{2} \right] \end{aligned} \end{aligned}$$
(D.5)

where the final equality follows from the reflection formula for the Gamma function

$$\begin{aligned} \Gamma (1 - z)\Gamma (z) = \frac{\pi }{\sin \pi z}. \end{aligned}$$

Taking \(c = \widehat{T}^{-1} \widehat{c}\) and sending \(\widehat{T} \rightarrow \infty \), we obtain (D.3).

Applying (D.1) again, we have

$$\begin{aligned}&\mathrm {Var}\left( \sum _{j=1}^\infty e^{c{\mathfrak {x}}_j^{(\widehat{T})}} \right) = {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \left( \sum _{j=1}^\infty e^{c{\mathfrak {x}}_j^{(\widehat{T})}} \right) ^2 \right] - {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \sum _{j=1}^\infty e^{c{\mathfrak {x}}_j^{(\widehat{T})}} \right] ^2 \\&\quad \quad = \frac{1}{(2\pi \mathbf {i})^2} \oint \oint \left( \frac{(u_1 - u_2)^2}{(u_1 - u_2 - c)(u_1 + c - u_2)} - 1 \right) \\&\quad \quad \prod _{i=1}^2 \frac{\Gamma (-u_i - c)}{\Gamma (-u_i)} \exp \left[ - \widehat{T} \left( \frac{c(c+1)}{2} + cu \right) \right] \frac{du_i}{c} \\&\quad \quad = \frac{1}{(2\pi \mathbf {i})^2} \oint \oint \frac{1}{(u_1 - u_2)^2 - c^2} \prod _{i=1}^2 \frac{\Gamma (-u_i - c)}{\Gamma (-u_i)} \exp \left[ - \widehat{T} \left( \frac{c(c+1)}{2} + cu \right) \right] \, du_i \end{aligned}$$

where the \(u_i\)-contour is positively oriented around \(-c,-c+1,-c+2,\ldots \) starting and ending at \(+\infty \) for \(i = 1,2\), and the \(u_2\)-contour encloses \(u_1 + c\) and \(u_1 - c\). Taking \(c = \widehat{T}^{-1} \widehat{c}\) and sending \(\widehat{T} \rightarrow \infty \), we obtain

$$\begin{aligned} \lim _{\widehat{T} \rightarrow \infty } \mathrm {Var}\left( \sum _{j=1}^\infty e^{\widehat{c} \widehat{T}^{-1} {\mathfrak {x}}_j^{(\widehat{T})}} \right) = \frac{1}{(2\pi \mathbf {i})^2} \oint \oint \frac{1}{(u_1 - u_2)^2} \prod _{i=1}^2 e^{-\widehat{c}(u_i + \frac{1}{2})} \, du_i \end{aligned}$$

where the \(u_i\)-contour is positively oriented around \({\mathbb {R}}_{\ge 0}\), starting and ending at \(+\infty \) for \(i = 1,2\), and the \(u_2\)-contour encloses the \(u_1\)-contour. Since the integrand decays as \(u_1 \rightarrow +\infty \) and is analytic as a function of \(u_1\) in the domain enclosed by the \(u_1\)-contour, the right hand side evaluates to 0. This proves (D.4). \(\square \)

1.2 D.2 \(\widehat{T} \rightarrow \infty \): Gaussian Fluctuations

We show that

$$\begin{aligned} \widehat{T}^{-1/2} \left( {\mathfrak {x}}_1^{(\widehat{T})} + \frac{\widehat{T}}{2} \right) \rightarrow \text{ standard } \text{ Gaussian } \end{aligned}$$
(D.6)

as \(\widehat{T} \rightarrow \infty \) using Laplace transforms. More precisely, we prove the following:

Theorem D.2

For \(0< \widehat{c} < 1\), we have

$$\begin{aligned} \lim _{\widehat{T} \rightarrow \infty } {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \sum _{j=1}^\infty e^{\widehat{c}\widehat{T}^{-1/2}\left( {\mathfrak {x}}_j^{(\widehat{T})} + \frac{\widehat{T}}{2} \right) } \right] = e^{\widehat{c}^2/2} \end{aligned}$$
(D.7)

and for \(0< \widehat{c}_1, \widehat{c}_2 < 1/2\),

$$\begin{aligned} \lim _{\widehat{T} \rightarrow \infty } {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^2 \sum _{j=1}^\infty e^{\widehat{c}_i \widehat{T}^{-1/2} \left( {\mathfrak {x}}_j^{(\widehat{T})} + \frac{\widehat{T}}{2} \right) } \right] = e^{(\widehat{c}_1 + \widehat{c}_2)^2/2}. \end{aligned}$$
(D.8)

We first show how Theorem D.2 implies (D.6):

Proof

(Proof of (D.6) using Theorem D.2) Let \(\rho ^{(\widehat{T})}_k\) denote the \(k\hbox {th}\) correlation function of \(\left\{ \widehat{T}^{-1/2}\left( {\mathfrak {x}}_i^{(\widehat{T})} + \tfrac{\widehat{T}}{2} \right) \right\} _{i=1}^\infty \). We first note that (D.7) implies

$$\begin{aligned} \rho ^{(\widehat{T})}_1(x) \, dx \rightarrow \frac{1}{\sqrt{2\pi }}e^{-x^2/2} \, dx \end{aligned}$$
(D.9)

weakly as \(\widehat{T} \rightarrow \infty \). The problem is that \(\rho ^{(\widehat{T})}_1\) is not quite the density of \(\widehat{T}^{-1/2}\left( {\mathfrak {x}}_1^{(\widehat{T})} + \tfrac{\widehat{T}}{2}\right) \), which we denote by \(p^{(\widehat{T})}\), and what we want is

$$\begin{aligned} p^{(\widehat{T})}(x) \, dx \rightarrow \frac{1}{\sqrt{2\pi }}e^{-x^2/2} \, dx \end{aligned}$$
(D.10)

weakly as \(\widehat{T} \rightarrow \infty \). For this, we need an argument which tells us that \(\rho ^{(\widehat{T})}_1\) is approximately given by \(p^{(\widehat{T})}\).

Observe that

$$\begin{aligned} \int _{{\mathbb {R}}^2} e^{\widehat{c}_1x_1 + \widehat{c}_2x_2}\rho _2^{(\widehat{T})}(x_1,x_2) \, dx_1 \, dx_2&= \left[ \sum _{j_1 \ne j_2} \prod _{i=1}^2 e^{\widehat{c}_i \widehat{T}^{-1/2} \left( {\mathfrak {x}}_{j_i}^{(\widehat{T})} + \frac{\widehat{T}}{2} \right) } \right] \\&= {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^2 \sum _{j=1}^\infty e^{\widehat{c}_i \widehat{T}^{-1/2} \left( {\mathfrak {x}}_j^{(\widehat{T})} + \frac{\widehat{T}}{2} \right) } \right] \\&\quad - {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \sum _{j=1}^\infty e^{(\widehat{c}_1 + \widehat{c}_2)\widehat{T}^{-1/2} \left( {\mathfrak {x}}_j^{(\widehat{T})} + \frac{\widehat{T}}{2} \right) } \right] \\&\rightarrow 0 \end{aligned}$$

as \(\widehat{T} \rightarrow \infty \), where the last line follows from Theorem D.2. As a consequence, we have

$$\begin{aligned} \lim _{\widehat{T} \rightarrow 0} \int _a^\infty \rho _2^{(\widehat{T})}(x_1,x_2) \, dx_1 \, dx_2 = 0 \end{aligned}$$

for any \(a \in {\mathbb {R}}\). Since

$$\begin{aligned} \int _a^\infty \rho _2^{(\widehat{T})}(x_1,x_2) \, dx_1 \, dx_2 = {{\,\mathrm{{\mathbb {E}}}\,}}\left[ N^{(\widehat{T})}(a) \left( N^{(\widehat{T})}(a) - 1\right) \right] \end{aligned}$$

where

$$\begin{aligned} N^{(\widehat{T})}(a) = \#\left\{ i: \widehat{T}^{-1/2} \left( {\mathfrak {x}}_j^{(\widehat{T})} + \tfrac{\widehat{T}}{2}\right) > a \right\} \end{aligned}$$

by e.g. [33, (1.2.2)  & (1.2.3)], we find that

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ N^{(\widehat{T})}(a)^2 \right] - {{\,\mathrm{{\mathbb {E}}}\,}}\left[ N^{(\widehat{T})}(a) \right] \rightarrow 0\end{aligned}$$

as \(\widehat{T} \rightarrow \infty \), and in particular

$$\begin{aligned} \lim _{\widehat{T} \rightarrow \infty } \P \left( N^{(\widehat{T})}(a) \le 1 \right) = 1. \end{aligned}$$

Therefore

$$\begin{aligned} \int _a^\infty \rho _1^{(\widehat{T})}(x) \, dx&= {{\,\mathrm{{\mathbb {E}}}\,}}\left[ N^{(\widehat{T})}(a) \right] \\&= {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \mathbf {1}\!\!\left[ \widehat{T}^{-1/2} \left( {\mathfrak {x}}_1^{(\widehat{T})} + \tfrac{\widehat{T}}{2} \right) > a \right] \right] + o(1) \\&= \int _a^\infty p^{(\widehat{T})}(x) \, dx + o(1) \end{aligned}$$

where the first equality uses [33, (1.2.2)  & (1.2.3)] again. Combining the above with (D.9), we obtain (D.10) as desired. \(\square \)

We now prove Theorem D.2.

Proof of Theorem D.2 We have the expansion (D.5) from the proof of Theorem D.1. Starting from there and taking \(c = \widehat{c} \widehat{T}^{-1/2}\), we have

$$\begin{aligned}&{{\,\mathrm{{\mathbb {E}}}\,}}\left[ \sum _{j=1}^\infty e^{\widehat{c} \widehat{T}^{-1/2}\left( {\mathfrak {x}}_j^{(\widehat{T})} + \frac{\widehat{T}}{2} \right) } \right] = \sum _{n=0}^\infty \frac{(-1)^n \sin ( \pi (\widehat{c} \widehat{T}^{-1/2} - n))}{\pi \widehat{c} \widehat{T}^{-1/2}} \frac{\Gamma (n+1-\widehat{c} \widehat{T}^{-1/2})}{\Gamma (n+1)} \\&\quad \exp \left[ \frac{\widehat{c}^2}{2} - \widehat{T}^{1/2} \widehat{c} n \right] . \end{aligned}$$

The \(n = 0\) term dominates the summation as \(\widehat{T} \rightarrow \infty \) and we obtain (D.9).

To prove (D.8), we obtain a similar residue expansion for (D.1) when \(m = 2\):

$$\begin{aligned}&{{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^2 \sum _{j=1}^\infty e^{c_i \left( {\mathfrak {x}}_j^{(\widehat{T})} + \frac{\widehat{T}}{2} \right) } \right] \\&\quad = \sum _{n=0}^\infty \frac{(-1)^n \sin (\pi (c_1 + c_2 - n))}{\pi (c_1 + c_2)} \frac{\Gamma (n + 1 - c_1 - c_2)}{\Gamma (n+1)} \exp \left[ - \widehat{T}(c_1 + c_2)n + \frac{\widehat{T}}{2} (c_1 + c_2)^2 \right] \\&\quad \quad + \sum _{n_1\ne n_2} \frac{(n_1 - n_2)(n_1 - c_1 - n_2 + c_2)}{(n_1 - n_2 + c_2)(n_1 - c_1 + n_2)} \prod _{i=1}^2 \frac{(-1)^{n_i} \sin (\pi (c_i - n_i))}{\pi c_i} \frac{\Gamma (n_i + 1 - c_i)}{\Gamma (n_i + 1)}\\&\quad \quad \exp \left[ -\widehat{T} c_i n_i + \widehat{T} \frac{c_i^2}{2} \right] \end{aligned}$$

where we have also applied the reflection formula for the Gamma function as in the proof of Theorem D.1. The second summation is over distinct \(n_1,n_2 \in \{0,1,2,\ldots \}\). Taking \(c_i = \widehat{c}_i \widehat{T}^{-1/2}\), the exponential terms dictate growth of each term as \(\widehat{T} \rightarrow \infty \). Thus we find that only the \(n = 0\) term in the first summation survives in the limit and (D.8) follows. \(\square \)

1.3 D.3 Airy point process

The process \(\zeta _1^{(\widehat{T})},\zeta _2^{(\widehat{T})},\ldots \) given by

$$\begin{aligned} {\mathfrak {x}}_i^{(\widehat{T})} = 2^{-1/3} \widehat{T}^{2/3} \zeta _i^{(\widehat{T})} + 1 + \log \widehat{T} \end{aligned}$$
(D.11)

converges to the Airy point process \({\mathfrak {a}}_1,{\mathfrak {a}}_2,\ldots \) as \(\widehat{T} \rightarrow 0\). In this sub-appendix we demonstrate, by way of a direct computation on Laplace transforms which is not fully rigorous, that

Theorem D.3

For \(\widehat{c}_1,\ldots ,\widehat{c}_m > 0\),

$$\begin{aligned} \lim _{\widehat{T} \rightarrow 0} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{\widehat{c}_i \zeta _j^{(\widehat{T})}} \right] = {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{\widehat{c}_i {\mathfrak {a}}_j} \right] . \end{aligned}$$

Proof Idea

From (D.11), it suffices to show

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{\widehat{c}_i 2^{1/3} \widehat{T}^{-2/3} {\mathfrak {x}}_i^{(\widehat{T})}} \right] = \left( e \widehat{T} \right) ^{2^{1/3} \widehat{T}^{-2/3} \sum _{i=1}^m \widehat{c}_i} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{\widehat{c}_i {\mathfrak {a}}_j} \right] (1 + o(1)) \end{aligned}$$
(D.12)

as \(\widehat{T} \rightarrow 0\). The Laplace transform for the correlation functions of the Airy point process can be computed explicitly, see [1, Appendix B], as

$$\begin{aligned} \begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{\widehat{c}_i {\mathfrak {a}}_j} \right] =&\frac{e^{\sum _{i=1}^m \widehat{c}_i^3/12}}{(2\pi \mathbf {i})^m} \int \frac{dz_1}{\widehat{c}_1} \cdots \int \frac{dz_m}{\widehat{c}_m} \\&\times \exp \left( \sum _{i=1}^m \widehat{c}_i z_i^2 \right) \prod _{1 \le i < j \le m} \frac{(z_j + \frac{\widehat{c}_j}{2} - z_i - \frac{\widehat{c}_i}{2})(z_j - \frac{\widehat{c}_j}{2} - z_i + \frac{\widehat{c}_i}{2})}{(z_j - \frac{\widehat{c}_j}{2} - z_i - \frac{\widehat{c}_i}{2})(z_j + \frac{\widehat{c}_j}{2} - z_i + \frac{\widehat{c}_i}{2})} \end{aligned} \end{aligned}$$
(D.13)

where the \(z_i\)-contour is a vertical contour from \(-\mathbf {i}\infty \) to \(\mathbf {i}\infty \) such that

$$\begin{aligned} \Re z_i + \frac{\widehat{c}_i}{2} < \Re z_j - \frac{\widehat{c}_j}{2} \end{aligned}$$

whenever \(1 \le i < j \le m\).

Therefore, we use the formula (D.1) with \(c_i = 2^{1/3} \widehat{T}^{-2/3} \widehat{c}_i\) and send \(\widehat{T} \rightarrow 0\) to see the desired asymptotics of (D.1) with the help of the explicit formula (D.13). The analysis of (D.1) relies on steepest descent which we outline. Set

$$\begin{aligned} F_{\widehat{T}}(v) := \log \Gamma (v) - \frac{\widehat{T}}{2} v^2. \end{aligned}$$

Then after changing variables \(v_i = -u_i\) in (D.1), we have

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{c_i {\mathfrak {x}}_i^{(\widehat{T)})}} \right] =&\frac{\prod _{i=1}^m e^{-\widehat{T} \frac{c_i}{2}}}{(2\pi \mathbf {i})^m} \oint \cdots \oint \det \left[ \frac{1}{v_i - v_j + c_j} \right] _{i,j=1}^m\nonumber \\&\prod _{i=1}^m e^{F_{\widehat{T}}(v_i - c_i) - F_{\widehat{T}}(v_i)} \, dv_i \end{aligned}$$
(D.14)

where the contour conditions for \(u_i\) translates to the condition that \(v_i - c_i\) and \(v_i + c_j\) are contained in the \(v_j\) contour for \(1 \le i < j \le m\). Observe that

$$\begin{aligned} f_{\widehat{T}}(v) := F_{\widehat{T}}'(v)&= \psi (v) - \widehat{T} v \end{aligned}$$

where \(\psi \) is the digamma function. The asymptotics below hold for \(|\arg v| < \pi - \delta \) below for some fixed \(\delta > 0\) (which we note does not correspond to the entire range of our contour integration). We have the asymptotic expansion

$$\begin{aligned} f_{\widehat{T}}(v) = -\widehat{T}v + \log v - \frac{1}{2v} - \frac{1}{12v^2} + O(1/|v|^4) \end{aligned}$$
(D.15)

as \(|v| \rightarrow \infty \), which can be differentiated arbitrarily many times for asymptotic expansions of the derivatives of \(f_{\widehat{T}}\). In particular, we find

$$\begin{aligned} f_{\widehat{T}}'''(v) = \psi '''(v) = \frac{2}{v^3} + O(1/|v|^4). \end{aligned}$$

Thus, if \(c \asymp T^{-2/3}\) and \(v \asymp \widehat{T}^{-1}\) as \(\widehat{T} \rightarrow 0\), we have

$$\begin{aligned} \begin{aligned} F_{\widehat{T}}(v - c) - F_{\widehat{T}}(v)&= -c f_{\widehat{T}}(v) + \frac{c^2}{2} f_{\widehat{T}}'(v) - \frac{c^3}{6} f_{\widehat{T}}''(v) + o(1) \end{aligned} \end{aligned}$$
(D.16)

as \(|v| \rightarrow \infty \), where we see that the first summand on the right hand side is positive for \(\widehat{T}\) large and is the leading order term from (D.15). Therefore, we want to localize the integral at a saddle point of the first term. Then we have a solution \(v_{\widehat{T}}\) to

$$\begin{aligned} f_{\widehat{T}}'(v_{\widehat{T}}) = \psi '(v_{\widehat{T}}) - \widehat{T} = 0 \end{aligned}$$

which satisfies

$$\begin{aligned} v_{\widehat{T}} = \widehat{T}^{-1} + O(1) \end{aligned}$$

from which we have

$$\begin{aligned} f_{\widehat{T}}(v_{\widehat{T}})&= - \log \widehat{T} - 1 + O(\widehat{T}) \\ f_{\widehat{T}}''(v_{\widehat{T}})&= - \widehat{T}^2 + O(\widehat{T}^3) \end{aligned}$$

using the expansion (D.15). If \(v - v_{\widehat{T}} = O(\widehat{T}^{-2/3 + \varepsilon })\) and \(c \asymp \widehat{T}^{-2/3}\) for \(\varepsilon > 0\) very small but fixed, then (D.16) implies

$$\begin{aligned} F_{\widehat{T}}(v - c) - F_{\widehat{T}}(v)&= -c \left( f_{\widehat{T}}(v_{\widehat{T}}) + \frac{1}{2} f_{\widehat{T}}''(v_{\widehat{T}})(v - v_{\widehat{T}})^2 \right) + \frac{c^2}{2} f_{\widehat{T}}''(v_{\widehat{T}}) (v - v_{\widehat{T}}) \\&\quad - \frac{c^3}{6} f_{\widehat{T}}''(v_{\widehat{T}}) + o(1) \\&= c\left( \log \widehat{T} + 1 \right) + \widehat{T}^2\left( \frac{c}{2} (v - v_{\widehat{T}})^2 - \frac{c^2}{2} (v - v_{\widehat{T}}) + \frac{c^3}{6} \right) + o(1) \end{aligned}$$

Writing \(c = 2^{1/3}\widehat{T}^{-2/3} \widehat{c}\) for \(\widehat{c} > 0\) fixed and \(z = 2^{-1/3} \widehat{T}^{2/3}(v - v_{\widehat{T}})\), we obtain

$$\begin{aligned} F_{\widehat{T}}(v - c) - F_{\widehat{T}}(v)&= 2^{1/3} \widehat{T}^{-2/3} \widehat{c} \left( \log \widehat{T} + 1 \right) + \widehat{c} z^2 - \widehat{c}^2 z + \frac{\widehat{c}^3}{3} + o(1) \\&= 2^{1/3} \widehat{T}^{-2/3} \widehat{c} \left( \log \widehat{T} + 1 \right) + \widehat{c}\left( z - \frac{\widehat{c}}{2}\right) ^2 + \frac{\widehat{c}^3}{12} + o(1). \end{aligned}$$

By choosing appropriate steepest descent contours, which we omit the details of here, we may localize the integral in (D.14) near \(v = v_{\widehat{T}}\). Using the asymptotics above, localizing the integrals near \(v_{\widehat{T}}\) (ignoring the details of justifying this localization), and changing variables \(z_i = 2^{-1/3} \widehat{T}^{2/3} (v_i - v_{\widehat{T}})\) in (D.14) and recalling \(c_i = 2^{1/3} \widehat{T}^{-2/3} \widehat{c}_i\), we obtain

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{c_i {\mathfrak {x}}_i^{(\widehat{T)})}} \right] =&\frac{1}{(2\pi \mathbf {i})^m} \int dz_1 \cdots \int dz_m \det \left[ \frac{1}{z_i - z_j + \widehat{c}_j} \right] _{i,j=1}^m \\&\quad \quad \times \prod _{i=1}^m \exp \left( 2^{1/3} \widehat{T}^{-2/3} \widehat{c}_i \left( \log \widehat{T} + 1 \right) + \widehat{c}_i \left( z_i - \frac{\widehat{c}_i}{2}\right) ^2 + \frac{\widehat{c}_i^3}{12} + o(1) \right) \\ =&\left( e \widehat{T} \right) ^{2^{1/3} \widehat{T}^{-2/3} \sum _{i=1}^m \widehat{c}_i} \frac{1}{(2\pi \mathbf {i})^m} \int dz_1 \cdots \int dz_m \\&\quad \quad \times \det \left[ \frac{1}{z_i - z_j + \widehat{c}_j} \right] _{i,j=1}^m \prod _{i=1}^m \exp \left( \widehat{c}_i \left( z_i - \frac{\widehat{c}_i}{2}\right) ^2 + \frac{\widehat{c}_i^3}{12}\right) (1 + o(1)) \end{aligned}$$

where the \(z_i\)-contours are vertical contours from \(-\mathbf {i}\infty \) to \(\mathbf {i}\infty \) such that

$$\begin{aligned} \Re z_i < \Re z_j - \widehat{c}_j \end{aligned}$$

whenever \(1 \le i < j \le m\). After changing variables so that \(z_i\) is replaced by \(z_i + \frac{\widehat{c}}{2}\), we recognize that the right hand side becomes

$$\begin{aligned} \left( e \widehat{T} \right) ^{2^{1/3} \widehat{T}^{-2/3} \sum _{i=1}^m \widehat{c}_i} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \prod _{i=1}^m \sum _{j=1}^\infty e^{\widehat{c}_i {\mathfrak {a}}_j} \right] (1 + o(1)) \end{aligned}$$

by (D.13), which yields (D.12). \(\square \)

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Ahn, A. Fluctuations of \(\beta \)-Jacobi product processes. Probab. Theory Relat. Fields 183, 57–123 (2022). https://doi.org/10.1007/s00440-022-01109-0

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Keywords

  • Products of random matrices
  • Singular values
  • Beta ensemble
  • Macdonald processes

Mathematics Subject Classification

  • 15B52
  • 60B20
  • 33D52