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Fluctuations for stationary q-TASEP

Abstract

We consider the q-totally asymmetric simple exclusion process (q-TASEP) in the stationary regime and study the fluctuation of the position of a particle. We first observe that the problem can be studied as a limiting case of an N-particle q-TASEP with a random initial condition and with particle dependent hopping rate. Then we explain how this N-particle q-TASEP can be encoded in a dynamics on a two-sided Gelfand–Tsetlin cone described by a two-sided q-Whittaker process and present a Fredholm determinant formula for the q-Laplace transform of the position of a particle. Two main ingredients in its derivation is the Ramanujan’s bilateral summation formula and the Cauchy determinant identity for the theta function with an extra parameter. Based on this we establish that the position of a particle obeys the universal stationary KPZ distribution (the Baik–Rains distribution) in the long time limit.

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Acknowledgements

Part of this work was performed during the stay at KITP in Santa Barbara, USA. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. The work of T.I. and T.S. is also supported by JSPS KAKENHI Grant Numbers JP25800215, JP16K05192 and JP25103004, JP14510499, JP15K05203, JP16H06338, JP18H01141, JP18H03672, respectively.

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Correspondence to Tomohiro Sasamoto.

Appendices

Appendix A: Some q-functions and q-formulas

In this appendix, we summarize a few q-notations, q-functions and q-formulas. The first is the q-Pochhammer symbol, or the q shifted factorial. For \(|q|<1\) and \(n\in \mathbb {N}\),

$$\begin{aligned} (a;q)_\infty = \prod _{n=0}^\infty (1-a q^n), ~~ (a;q)_n = \frac{(a;q)_\infty }{(a q^n;q)_\infty }. \end{aligned}$$
(A.1)

The q-binomial theorem will be useful in various places in the discussions,

$$\begin{aligned} \sum _{n=0}^\infty \frac{(a;q)_n}{(q;q)_n} z^n = \frac{(az;q)_\infty }{(z;q)_\infty }, ~ |z|<1. \end{aligned}$$
(A.2)

In particular the \(a=0\) case appears in many applications. Another q-binomial formula reads (see e.g. Cor.10.2.2.(b) in [6])

$$\begin{aligned} \sum _{n=0}^\infty \frac{(-1)^n q^{n(n-1)/2}}{(q;q)_n} z^n = (z;q)_\infty . \end{aligned}$$
(A.3)

There is yet another version of the q-binomial theorem (see e.g. Cor.10.2.2.(c) in [6]),

$$\begin{aligned} \sum _{k=0}^{\ell } \frac{(-1)^kq^{k(k-1)/2}(q;q)_{\ell }}{(q;q)_k(q;q)_{\ell -k}}x^k = (1-x)(1-xq)\cdots (1-xq^{\ell -1}). \end{aligned}$$
(A.4)

The q-exponential function, denoted as \(e_q(z)\) is defined to be

$$\begin{aligned} e_q(z) := \frac{1}{((1-q)z;q)_\infty } = \sum _{n=0}^\infty \frac{(1-q)^n}{(q;q)_n} z^n. \end{aligned}$$
(A.5)

The second equality is by the above q-binomial theorem (A.2). From the series expansion expression, it is easy to see that this tends to the usual exponential function in the \(q\rightarrow 1\) limit.

The q-Gamma function \(\Gamma _q(x)\) is defined by

$$\begin{aligned} \Gamma _q(x) = (1-q)^{1-x} \frac{(q;q)_{\infty }}{(q^x;q)_{\infty }}. \end{aligned}$$
(A.6)

The q-digamma function is defined by \(\Phi _q(z)=\partial _z\log \Gamma _q(z)\). In the \(q\rightarrow 1\) limit, they tends to the usual \(\Gamma \) function and the digamma function respectively.

Ramanujan’s summation formula (cf [6, p502], [38, p138]) is a two-sided generalization of the above q-binomial theorem (A.2). For \(|q|<1, |b/a|<|z|<1\),

$$\begin{aligned} \sum _{n\in \mathbb {Z}} \frac{(a;q)_n}{(b;q)_n} z^n = \frac{(az;q)_\infty \left( \frac{q}{az};q\right) _\infty (q;q)_\infty \left( \frac{b}{a};q\right) _\infty }{(z;q)_\infty \left( \frac{q}{a};q\right) _\infty (b;q)_\infty \left( \frac{b}{az};q\right) _\infty } . \end{aligned}$$
(A.7)

Appendix B: q-Whittaker functions and q-Whittaker process

In this appendix, we introduce the ordinary (skew) q-Whittaker functions labeled by partitions and the q-Whittaker process and then discuss some of their properties. Most of them are standard [61] but some of them are reformulated for the applications in the main text.

First a partition of length \(n\in \mathbb {N}\) is an n-tuple \(\lambda =(\lambda _1,\ldots ,\lambda _n)\) with \(\lambda _j\in \mathbb {Z}_+, 1\le j\le n\) s.t. \(\lambda _1\ge \ldots \ge \lambda _n\). The set of all partitions of length n is denoted by \(\mathcal {P}_n\). A partition \(\lambda \) can also be represented (and identified) as a Young diagram with n rows of length \(\lambda _1,\ldots ,\lambda _n\). For two partitions \(\lambda \in \mathcal {P}_n,\mu \in \mathcal {P}_m\) s.t. \(m\le n\) and \(\lambda _i-\mu _i\ge 0, 1\le i\le n\) (with the understanding \(\mu _i\equiv 0, m<i\le n)\), a pair \((\lambda ,\mu )\) is called a skew diagram and is denoted by \(\lambda /\mu \). The transpose \(\lambda '\) is the partition of length \(\lambda _1\) defined as \(\lambda _i'=\#\{j\in \mathbb {Z}_+ | \lambda _j\ge i\},1\le i\le \lambda _1\).

The Gelfand–Tsetlin (GT) cone for partitions, denoted by \(\mathbb {G}_N^{(0)},N\in \mathbb {N}\) here, is defined by

$$\begin{aligned} \mathbb {G}^{(0)}_N :=&\, \{ (\lambda ^{(1)},\lambda ^{(2)},\ldots ,\lambda ^{(N)}), \lambda ^{(n)}\in \mathcal {P}_n,1\le n\le N| \nonumber \\ 0\le&\, \lambda ^{(m+1)}_{\ell +1}\le \lambda ^{(m)}_{\ell }\le \lambda ^{(m+1)}_{\ell }, 1\le \ell \le m \le N-1 \}. \end{aligned}$$
(B.1)

An element \(\underline{\lambda }_N\in \mathbb {G}_N^{(0)}\) is called a Gelfand–Tsetlin pattern. Next we define the (skew) q-Whittaker functions.

Definition B.1

Let \(\lambda \in \mathcal {P}_n,\mu \in \mathcal {P}_{n-1}\) be two partitions of order n and \(n-1\) respectively and a an indeterminate. The skew q-Whittaker function (with 1 variable) is defined as ([61, VI.7, (7.14) and Ex. 2.])

$$\begin{aligned} P_{\lambda /\mu }\left( a \right) =\prod _{i=1}^n a^{\lambda _i}\cdot \prod _{i=1}^{n-1}\frac{ a^{-\mu _i} (q;q)_{\lambda _i-\lambda _{i+1}}}{(q;q)_{\lambda _i-\mu _i}(q;q)_{\mu _i-\lambda _{i+1}}} . \end{aligned}$$
(B.2)

Using this, for a partition \(\lambda \in \mathcal {P}_N\) and \(a=(a_1,\ldots ,a_N)\) being N indeterminates, we define the q-Whittaker function with N variables as (cf. [61, VI.7, (7.9’)])

$$\begin{aligned} P_\lambda \left( a\right) = \sum _{\begin{array}{c} \lambda _i^{(k)}, 1\le i\le k\le N-1\\ \lambda _{i+1}^{(k+1)} \le \lambda _i^{(k)} \le \lambda _i^{(k+1)} \end{array}} \prod _{j=1}^N P_{\lambda ^{(j)}/\lambda ^{(j-1)}}\left( a_j\right) . \end{aligned}$$
(B.3)

Here the sum is over the Gelfand–Tsetlin cone \(\mathbb {G}_N^{(0)}\) with the condition \(\lambda ^{(N)}=\lambda \) and \(\lambda ^{(0)}=\phi \).

It is known that the q-Whittaker function \(P_\lambda (a)\) forms a basis of \(\Lambda _N\), the space of N-variable symmetric polynomials with coefficients being rational functions in q (cf [61, VI]). There is an inner product \(\langle ~,~ \rangle \) in this space for which \(P_\lambda \)’s are orthogonal ([61, VI (6.19)]):

$$\begin{aligned} \langle P_\lambda ,P_\mu \rangle =(q;q)_{\lambda _N} \prod _{i=1}^{N-1}(q;q)_{\lambda _{i}-\lambda _{i+1}} \delta _{\lambda ,\mu }. \end{aligned}$$
(B.4)

Using this, we also introduce \(Q_\lambda (x)\).

Definition B.2

For \(\lambda \in \mathcal {P}_N\) and for \(x=(x_1,\ldots ,x_N)\), we define ([61, VI (4.11), (4.12)])

$$\begin{aligned} Q_\lambda (x)&= \frac{P_\lambda (x)}{\langle P_\lambda ,P_\lambda \rangle }. \end{aligned}$$
(B.5)

Note that \(P_\lambda \) and \(Q_\lambda \) are orthonormal: \(\langle P_\lambda , Q_\mu \rangle = \delta _{\lambda ,\mu }\).

If we take a sum over all partitions of length N of a product of \(P_\lambda \) and \(Q_\lambda \), the following Cauchy identity for the q-Whittaker functions holds [61, VI(4.13), (2.5)],

$$\begin{aligned} \sum _{\lambda \in \mathcal {P}_N} P_{\lambda }(x) Q_{\lambda }(y)=\prod _{i,j=1}^N\frac{1}{(x_i y_j;q)_{\infty }} =:\Pi (x;y). \end{aligned}$$
(B.6)

Let us rewrite this identity in a form used in the main text. First one notices that one can express \(P_{\lambda }(x)\) in a form,

$$\begin{aligned} P_\lambda (x)=X^{\lambda _N}R_\ell (x), \end{aligned}$$
(B.7)

where \(X=x_1 \ldots x_N\) and \(\ell =(\ell _1,\ldots ,\ell _{N-1})\) with \(\ell _j=\lambda _{j}-\lambda _{j+1}, 1\le j\le N-1\) and \(R_\ell \in \Lambda _N\). This is seen as follows. Since \((\lambda ^{(1)},\ldots , \lambda ^{(N)})\) in (B.3) is an element of \(\mathbb {G}_N\), the order of each term in the sum in (B.3) is bigger than or equal to \(\lambda ^{(N)}_N\) and one can factor out this lowest order common factor. Then a partition \(\lambda \) is uniquely determined by \(\lambda ^{(N)}_N\) and \(\ell _j,j=1,...,N-1\) and the coefficient of each term depends only on \(\ell _j,j=1,\ldots ,N-1\). Then we have (B.7). We do not write explicitly the form of the function \(R_\ell (x)\) since it is not necessary in the following discussion. In terms of \(R_\ell \), the Cauchy identity (B.6) is equivalent to

$$\begin{aligned} \sum _{\ell _1,\ldots ,\ell _{N-1}=0}^{\infty } R_\ell (x)R_\ell (y) \prod _{j=1}^{N-1} \frac{1}{(q;q)_{\ell _j}} = \frac{\left( XY ;q\right) _{\infty }}{\prod _{i,j=1}^N( x_i y_j;q)_{\infty }} \end{aligned}$$
(B.8)

with \(Y=y_1 \ldots y_N\). In fact if one substitutes (B.7) into lhs of (B.6) and uses (B.8) and the q-binomial theorem (A.2) with \(a=0\), we get rhs of (B.6).

There is another inner product \(\langle \cdot ,\cdot \rangle '\) in \(\Lambda _N\) called the torus scaler product: For the N-variable functions, \(f(z), g(z)\in \Lambda _N\), the torus scalar product is defined by [61, VI (9.10)]

$$\begin{aligned} \langle f,g\rangle '_N&= \int _{\mathbb {T}^N}\prod _{j=1}^N \frac{dz_j}{z_j} \cdot f(z)\overline{g(z)} m_N^q(z), \end{aligned}$$
(B.9)

where

$$\begin{aligned} m_N^q(z)=\frac{1}{(2\pi i)^NN!}\prod _{1\le i<j\le N}(z_i/z_j;q)_{\infty }(z_j/z_i;q)_{\infty } \end{aligned}$$
(B.10)

is the q-Sklyanin measure. The q-Whittaker functions of N variables are known to satisfy the following orthogonality relations with respect to this inner product  [61],

$$\begin{aligned} \langle P_{\lambda },P_{\mu }\rangle '_N = \prod _{i=1}^N \frac{1}{\left( q^{\lambda _i-\lambda _{i+1}+1};q\right) _{\infty }} \cdot \delta _{\lambda ,\mu }. \end{aligned}$$
(B.11)

Here we rewrite the orthogonality relation (B.11). From the definition (B.3), we find that \(P_{\lambda }(e^{i\theta })=P_{\lambda }(e^{i\theta _1},\ldots ,e^{i\theta _N})\) can be expressed as

$$\begin{aligned} P_{\lambda }(e^{i\theta })=e^{iN\bar{\theta }\bar{\lambda }}\tilde{P}_{\ell }(\tilde{\theta }), \end{aligned}$$
(B.12)

where \(\bar{\theta }=(\theta _1+\cdots +\theta _N)/N\) and \(\bar{\lambda }=(\lambda _1+\cdots +\lambda _N)/N\) represent the barycentric coordinates while relative coordinates are denoted by \(\tilde{\theta }=(\tilde{\theta }_1,\ldots ,\tilde{\theta }_{N-1})\) and \(\ell =(\ell _1,\ldots ,\ell _{N-1})\) with \(\tilde{\theta }_j=\theta _j-\theta _{j+1}\) and \(\ell _j=\lambda _j-\lambda _{j+1}, 1\le j\le N-1\). Considering this we see that (B.11) can be rewritten as, with \(m_j=\mu _j-\mu _{j+1}, 1\le j\le N-1\)

$$\begin{aligned} \langle P_{\lambda },P_{\mu }\rangle '_N&= \int _{0}^{2\pi }d\bar{\theta } e^{iN\bar{\theta }(\bar{\lambda }-\bar{\mu })} \int _{(-\pi ,\pi ]^{N-1}}d\tilde{\theta } \tilde{P}_{\ell }(\tilde{\theta })\tilde{P}_{m}(-\tilde{\theta }) \prod _{1\le j<k\le N-1} \left| (e^{i(\tilde{\theta }_j+\cdots +\tilde{\theta }_k)};q)_{\infty }) \right| ^2\nonumber \\&= \prod _{i=1}^N \frac{1}{\left( q^{\ell _i+1};q\right) _{\infty }} \cdot \delta _{\lambda ,\mu }. \end{aligned}$$
(B.13)

Hence for the difference variables \(\ell ,m\in \mathbb {Z}_{\ge 0}^{N-1}\), the orthogonality relation (B.11) can be restated as

$$\begin{aligned} \int _{(-\pi ,\pi ]^{N-1}}d\tilde{\theta } \tilde{P}_{l}(\tilde{\theta })\tilde{P}_{m}(-\tilde{\theta }) \prod _{1\le j<k\le N-1} \left| (e^{i(\tilde{\theta }_j+\cdots +\tilde{\theta }_k)};q)_{\infty }) \right| ^2 = \prod _{i=1}^N \frac{1}{\left( q^{l_i+1};q\right) _{\infty }} \cdot \delta _{l,m}. \end{aligned}$$
(B.14)

One finds a representation of \(Q_\lambda \) using torus scalar product,

Lemma B.3

For \(y\in \mathbb {R}^N,\lambda \in \mathcal {P}_N\),

$$\begin{aligned} Q_\lambda \left( y\right)&=\frac{1}{\langle P_\lambda ,P_\lambda \rangle '_N } \langle \Pi (\cdot ,y),P_\lambda (\cdot )\rangle _N'\nonumber \\&=\prod _{i=1}^{N-1}\left( q^{\lambda _i-\lambda _{i+1}+1};q\right) _{\infty } \int _{\mathbb {T}^N}\prod _{i=1}^N\frac{dz_i}{z_i}\cdot P_\lambda \left( 1/z\right) \Pi \left( z;y\right) m_N^q\left( z\right) , \end{aligned}$$
(B.15)

where 1 / z in \(P_\lambda \) is a shorthand notation for \((1/z_1,\ldots ,1/z_N)\).

Proof

It immediately follows from the Cauchy identity (B.6) and the orthogonality (B.11). \(\square \)

We can consider a generalization of the function \(Q_\lambda \) by modifying \(\Pi (z;y)\) in (B.15) to

$$\begin{aligned} \Pi (z;\{\alpha ,\beta ,\gamma \}) = \prod _{i=1}^N\prod _{j=1}^M \frac{1+z_i \beta _j}{(z_i \alpha _j;q)_\infty } e^{\gamma z_i} \end{aligned}$$
(B.16)

with \(\alpha _j,\beta _j, \gamma \ge 0\) for \(j=1,\ldots ,M\). We call this \(Q_{\lambda } \left( \{\alpha ,\beta ,\gamma \}\right) \). The case where only some of \(\alpha _j\)’s (resp. \(\beta _j\)’s or \(\gamma \)) are positive is called the \(\alpha \)-specialization (resp. \(\beta \)-specialization or Plancherel specialization) in [16]. Note that the (B.16) for the \(\alpha \)-specialization is nothing but \(\Pi (z;\alpha )\) in (B.6). The (ascending) q-Whittaker process corresponding to this generalization is defined by

$$\begin{aligned} P (\underline{\lambda }_N) := \frac{\prod _{j=1}^N P_{\lambda ^{(j)}/\lambda ^{(j-1)}} \left( a_j\right) \cdot Q_{\lambda ^{(N)}} \left( \{\alpha ,\beta ,\gamma \} \right) }{\Pi (z;\{\alpha ,\beta ,\gamma \})}. \end{aligned}$$
(B.17)

For instance we define \(Q^{(\beta )}_{\lambda }(y),y\in \mathbb {R}^M,M\in \mathbb {N}\) by (B.15) with \(\Pi (z;y)\) replaced by \(\Pi _M^{(\beta )}(z;y):=\Pi (z;{\alpha =0,\beta =y,\gamma =0})= \prod _{i=1}^N\prod _{j=1}^M(1+z_i y_j)\). We state two properties of the q-Whittaker functions with this \(\beta \)-specialization, which will be also useful to discuss our q-TASEP with a random initial condition (cf [61, VI 7]).

$$\begin{aligned}&\sum _{\kappa }P_{\kappa /\lambda }(a)Q^{(\beta )}_{\kappa /\mu }(r) = \sum _{\tau }P_{\mu /\tau }(a)Q^{(\beta )}_{\lambda /\tau }(r)\cdot (1+ra), \end{aligned}$$
(B.18)
$$\begin{aligned}&\sum _{\mu }Q_{\lambda /\mu }^{(\beta )}(r_{M+1}) Q_{\mu }^{(\beta )}(\{r\}_M) = Q_{\lambda }^{(\beta )}(\{r\}_{M+1}), \end{aligned}$$
(B.19)

where in (B.19), \(Q^{(\beta )}_{\lambda /\mu }(r)\) with one variable is defined as

$$\begin{aligned} Q^{(\beta )}_{\lambda /\mu }(r)=\prod _{i\ge 1: \lambda _i=\mu _i,\lambda _{i+1}=\mu _{i+1}+1} \left( 1-q^{\mu _i-\mu _{i+1}}\right) r^{|\lambda |-|\mu |} \end{aligned}$$
(B.20)

and we introduced a notation \(\{r\}_M=(r_1,\ldots ,r_M)\) to consider a change of M below.

In the rest of this appendix we consider some properties of the q-Whittaker process corresponding to this pure \(\beta \) specialization.

Definition B.4

For \(M\in \mathbb {N}\),

$$\begin{aligned} P_M(\underline{\lambda }_N) = \frac{1}{\Pi ^{(\beta )}(a;r)}\prod _{j=1}^NP_{\lambda ^{(j)}/\lambda ^{(j-1)}}(a_j)\cdot Q_{\lambda ^{(N)}}^{(\beta )}(\{r\}_M). \end{aligned}$$
(B.21)

As mentioned above, the positivity of (B.21) is known. This process describes a distribution function at time M of a discrete time Markov process on \(\mathbb {G}_N^{(0)}\) described by the following Kolmogorov forward equation.

Proposition B.5

$$\begin{aligned} P_{M+1}(\underline{\lambda }_N) = \sum _{\underline{\mu }_N} P_{M}(\underline{\mu }_N)G_{M}(\underline{\mu }_N,\underline{\lambda }_N) \end{aligned}$$
(B.22)

where the transition matrix \(G_M(\underline{\mu }_N,\underline{\lambda }_N)\) is

$$\begin{aligned}&G_M(\underline{\mu }_N,\underline{\lambda }_N) = \prod _{j=1}^N \frac{P_{\lambda ^{(j)}/\lambda ^{(j-1)}}(a_j)Q^{(\beta )}_{\lambda ^{(j)}/\mu ^{(j)}}(r_{M+1})}{\Delta (\lambda ^{(j-1)},\mu ^{(j)})}, \end{aligned}$$
(B.23)

with \(\Delta (\lambda ,\mu )=\sum _{\kappa }P_{\kappa /\lambda }(a)Q^{(\beta )}_{\kappa / \mu }(r_M)\).

Remark

It is easy to see that each factor in (B.23) is a transition probability matrix from the definition of \(\Delta (\lambda ,\mu )\).

Proof

We rewrite \(G_{M}(\underline{\mu }_N,\underline{\lambda }_N)\) as

$$\begin{aligned} G_{M}(\underline{\mu }_N,\underline{\lambda }_N) = \prod _{r=1}^N A^{(\beta )}_r(r_{M+1})\cdot B^{(\beta )}(r_{M+1})\prod _{j=1}^N P_{\lambda ^{(j)}/\lambda ^{(j-1)}}(a_j), \end{aligned}$$
(B.24)

where

$$\begin{aligned} A^{(\beta )}_r(r_{M+1}) = \frac{Q^{(\beta )}_{\lambda ^{(r-1)}/\mu ^{(r-1)}}(r_{M+1})}{\Delta (\lambda ^{(r-1)},\mu ^{(r)})},~ B^{(\beta )}(r_{M+1})=Q^{(\beta )}_{\lambda ^{(N)}/\mu ^{(N)}}(r_{M+1}). \end{aligned}$$
(B.25)

Using (B.18) and (B.19), we have

$$\begin{aligned}&\sum _{\mu ^{(r-1)}}P_{\mu ^{(r)}/\mu ^{(r-1)}}(a_r) A_{r}^{(\beta )}(r_{M+1})=\frac{1}{1+a_r r_{M+1}},~1\le r\le N, \end{aligned}$$
(B.26)
$$\begin{aligned}&\sum _{\mu ^{(N)}}Q^{(\beta )}_{\mu ^{(N)}}(\{r\}_{M})B^{(\beta )}(r_{M+1}) =Q_{\lambda ^{(N)}}^{(\beta )}(\{r\}_{M+1}). \end{aligned}$$
(B.27)

Thus we have

$$\begin{aligned}&\sum _{\underline{\mu }_N} P_{M}(\underline{\mu }_N)G_M(\underline{\mu }_N,\underline{\lambda }_N)\nonumber \\&\quad = \frac{1}{\Pi _M^{(\beta )}(a;r)} \prod _{r=1}^N \left( \sum _{\mu ^{(r-1)}}P_{\mu ^{(r)}/\mu ^{(r-1)}}(a_r) A_{r}^{(\beta )}(r_{M+1})\right) \nonumber \\&\qquad \times \sum _{\mu ^{(N)}}Q^{(\beta )}_{\mu ^{(N)}} (\{r\}_M)B^{(\beta )}(r_{M+1}) \cdot \prod _{j=1}^N P_{\lambda ^{(j)}/\lambda ^{(j-1)}}(a_j) \nonumber \\&\quad = \frac{1}{\Pi _M^{(\beta )}(a;r)} \prod _{r=1}^N \frac{1}{1+a_r r_{M+1}} \cdot Q_{\lambda ^{(N)}}^{(\beta )}(\{r\}_{M+1}) \cdot \prod _{j=1}^N P_{\lambda ^{(j)}/\lambda ^{(j-1)}}(a_j) = P_{M+1}(\underline{\lambda }_N). \end{aligned}$$
(B.28)

\(\square \)

Appendix C: Two lemmas regarding the Airy kernel and the GUE Tracy–Widom limit

Here we provide two lemmas for establishing the GUE Tracy–Widom limit when a kernel of a specific form is given. The Hermite kernel for the GUE is a simplest example. For GUE, one can also use a bound due to Ledoux, which holds for GUE and simplifies the proof for the case (cf [4, 57]), but in other applications such a bound is not available. Our lemmas do not rely on such extra information but focus only on properties of the kernel. The essential part of the arguments are given in [21, 49] but we reformulate and generalize them in a way which would be suited for various applications. The Airy kernel \(K(\xi ,\zeta )\) is defined as

$$\begin{aligned} K(\xi ,\zeta ) = \int _0^\infty d\lambda \mathrm{Ai}(\xi +\lambda ) \mathrm{Ai}(\zeta +\lambda ). \end{aligned}$$
(C.1)

Our discussions below are given in a continuous setting but can be also applied to a discrete setting as well.

Lemma C.1

Suppose that we have a kernel of the form,

$$\begin{aligned} K_N^{(0)}(x,y) = \sum _{n=0}^{N-1} \varphi _n(x) \psi _n(y), \end{aligned}$$
(C.2)

where \(\varphi _n,\psi _n,n\in \mathbb {N}\) are complex functions on \(\mathbb {R}\). Assume that the functions \(\varphi _n(x)\) satisfy the followings for some \(a\in \mathbb {R}, \gamma>0, c>0\).

  1. (a)

    For \(\forall M>0\) and \(\forall L>0\),

    $$\begin{aligned} \lim _{N\rightarrow \infty } \gamma _1N^{1/3} \varphi _{N-cN^{1/3}\lambda }(aN+\gamma N^{1/3}\xi ) = \mathrm{Ai}(\xi +\lambda ), \end{aligned}$$
    (C.3)

    uniformly for \(|\xi |<L\) and for \(\lambda \in [0,M]\).

  2. (b)

    For \(\forall L>0, \forall \epsilon >0\) and N large enough,

    $$\begin{aligned} \gamma _1N^{1/3} \varphi _{N-cN^{1/3}\lambda }(aN+\gamma N^{1/3}\xi ) \le c_{b} e^{-(\xi +\lambda )}, \end{aligned}$$
    (C.4)

    for \(\xi ,\lambda \) satisfying \(\lambda >0, L\le |\xi +\lambda | \le \epsilon N^{2/3}\) and for some constant \(c_b\).

  3. (c)

    For \(\forall \epsilon >0\) and N large enough,

    $$\begin{aligned} \gamma _1N^{1/3} \varphi _{N-cN^{1/3}\lambda }(aN+\gamma N^{1/3}\xi ) \le c_{c} e^{-(\xi +\lambda )} , \end{aligned}$$
    (C.5)

    for \(\xi ,\lambda \) satisfying \(\lambda>0, |\xi +\lambda |> \epsilon N^{2/3}\) and for some constants \(c_c\).

The functions \(\psi _n\) are also assumed to satisfy the same conditions (a), (b), (c) with the parameters \(a\in \mathbb {R}, c>0\), same as above and \(\gamma _2>0\). Define the rescaled kernel, with \(\gamma _1\gamma _2=c\gamma \),

$$\begin{aligned} K_N(\xi ,\zeta ) = \gamma N^{1/3} K_N^{(0)}(aN+\gamma N^{1/3}\xi ,aN+\gamma N^{1/3}\zeta ). \end{aligned}$$
(C.6)

Then we have

  1. (i)

    For \(\forall L>0\),

    $$\begin{aligned} \lim _{N\rightarrow \infty } K_N(\xi ,\zeta ) = K(\xi ,\zeta ), \quad \text {uniformly (in}\ N)\ \text {on} \quad [-L,L]^2. \end{aligned}$$
    (C.7)
  2. (ii)

    For \(\forall L>0\),

    $$\begin{aligned} {|} K_N(\xi ,\zeta )| \le c_1 e^{-\max (0,\xi )-\max (0,\zeta )} , \quad \text {uniformly (in}\ N\text {) for} \quad \xi ,\zeta \ge -L \end{aligned}$$
    (C.8)

    for some constant \(c_1\).

Proof

First we divide the sum over n in (C.2) as \(\sum _{n=0}^{N-1} = \sum _{n=0}^{[N-cN^{1/3}M]} + \sum _{n=[N-cN^{1/3}M]+1}^{N-1}\), where [x] is the maximum integer which is less than or equal to \(x\in \mathbb {R}\). For the second sum, we have, due to the uniform convergence in (a),

$$\begin{aligned}&\lim _{N\rightarrow \infty } \frac{1}{cN^{1/3}}\sum _{n=N-cN^{1/3}M}^{N-1} \gamma _1N^{1/3}\varphi _n(aN+c N^{1/3}\xi ) \gamma _1N^{1/3}\psi _n(aN+c N^{1/3}\zeta ) \nonumber \\&\quad = \int _0^M d\lambda \mathrm{Ai}(\xi +\lambda ) \mathrm{Ai}(\zeta +\lambda ) \end{aligned}$$
(C.9)

for \(\xi ,\zeta \in [-L,L]\) for \(\forall L>0\). The first sum is, due to (c), (d), bounded by \(\int _M^\infty e^{-2\lambda } d\lambda = e^{-2M}/2\). By taking the \(M\rightarrow \infty \) limit, we have (C.7) in (i). The bound in (ii) easily follows from the uniform convergence in (a) (Note \(\mathrm{Ai}(\xi )\) is bounded by a constant as \(\xi \rightarrow -\infty \) and by \(e^{-\xi }\) as \(\xi \rightarrow \infty \)) combined with the bounds in (b), (c). \(\square \)

Lemma C.2

Suppose that a sequence of kernel \(K_N: \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {C}, N\in \mathbb {N}\) satisfies

  1. (i)

    For \(\forall L>0\),

    $$\begin{aligned} \lim _{N\rightarrow \infty } K_N(\xi ,\zeta ) = K(\xi ,\zeta ), \quad \text {uniformly in} \quad [-L,L]^2. \end{aligned}$$
    (C.10)
  2. (ii)

    For \(\forall L>0\),

    $$\begin{aligned} {|} K_N(\xi ,\zeta )| \le c_1 e^{-\max (0,\xi )-\max (0,\zeta )} , \quad \text {uniformly (in}\ N\text {) for} \quad \xi ,\zeta \ge -L, \end{aligned}$$
    (C.11)

    for some constant \(c_1\).

    In addition suppose that a (sequence of ) function \(f_N: \mathbb {R}\rightarrow \mathbb {R},N\in \mathbb {N}\) satisfy

  3. (iii)

    The functions \(f_N,N\in \mathbb {N}\) are uniformly bounded and converges to \(1_{(s,\infty )}\) for a \(s\in \mathbb {R}\) in the \(L^1(\mathbb {R})\) norm,

    $$\begin{aligned} \lim _{N\rightarrow \infty } \int _\mathbb {R}|f_N (\xi ) - 1_{(s,\infty )} (\xi )|d\xi =0. \end{aligned}$$
    (C.12)
  4. (iv)

    For \(\forall L(>|s|)\),

    $$\begin{aligned} \sqrt{|f_N(\xi )f_N(\zeta )|} |K_N(\xi ,\zeta )| \le e^{\xi +\zeta } \end{aligned}$$
    (C.13)

    and

    $$\begin{aligned} \lim _{N\rightarrow \infty } \sqrt{|f_N(\xi )f_N(\zeta )|} |K_N(\xi ,\zeta )| =0 \end{aligned}$$
    (C.14)

    uniformly (in N) for \(\xi ,\zeta \) satisfying \(\xi \le -L\) or \(\zeta \le -L\).

    Then for \(\forall s\in \mathbb {R}\),

    $$\begin{aligned} \lim _{N\rightarrow \infty } \det (1-f_N K_N)_{L^2(\mathbb {R})} = F_2(s). \end{aligned}$$
    (C.15)

Remark

The condition (i), (ii) are the same as (i)(ii) in Proposition C.1. For \(f_N= 1_{(s,\infty )},\forall N\in \mathbb {N}\), (iii)(iv) are trivially satisfied. Hence the kernel \(K_N\) from Proposition C.1 with this special \(f_N\) automatically satisfies the above four conditions. This special case appears in many applications, for example in GUE and TASEP.

Proof

In this proof \(c_i,i=1,2,\ldots \) are some constants. We first prove the \(f_N= 1_{(s,\infty )},\forall N\in \mathbb {N}\) case. By the remark above, we will show that for a kernel satisfying (i)(ii) in Proposition C.1 we have

$$\begin{aligned} \lim _{N\rightarrow \infty } \det (1-K_N)_{L^2(s,\infty )} = F_2(s). \end{aligned}$$
(C.16)

The general case will be treated by considering the difference to this special case later in the proof. For a given \(s\in \mathbb {R}\), take \(L(>|s|)\). By the Hadamard’s inequality,

$$\begin{aligned} {|}\det A | \le \prod _{i=1}^n \left( \sum _{j=1}^n |a_{i,j}|^2\right) ^{1/2}, \end{aligned}$$
(C.17)

which holds for general \(n\times n\) matrix [41], and (ii), we have for \(\xi ,\zeta \ge -L\),

$$\begin{aligned} {|}\det (K_N(\xi _i,\xi _j))_{1\le i,j\le k}|&\le \prod _{i=1}^k \left( \sum _{j=1}^k |K_N(\xi _i,\xi _j)|^2 \right) ^{1/2} \nonumber \\&\le \prod _{i=1}^k \left( \sum _{j=1}^k c_1^2( e^{-\max (0,\xi _i)-\max (0,\xi _j)} )^2\right) ^{1/2} \nonumber \\&\le c_1^k k^{k/2} \prod _{i=1}^k e^{-\max (0,\xi _i)}, \end{aligned}$$
(C.18)

and hence

$$\begin{aligned} \left| \int _{(s,\infty )^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \le c_1^k k^{k/2} \prod _{i=1}^k \int _s^{\infty } e^{-\max (0,\xi _i)} d\xi _i \le c_2^k k^{k/2}. \end{aligned}$$
(C.19)

Take \(\epsilon >0\). By (C.19), the Fredholm expansion,

$$\begin{aligned} \det (1-K_N)_{L^2(s,\infty )} = \sum _{k=0}^\infty \frac{(-1)^k}{k!} \int _{(s,\infty )^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i, \end{aligned}$$
(C.20)

converges and there exists \(l\in \mathbb {N}\) s.t.

$$\begin{aligned}&\left| \det (1-K_N)_{L^2(s,\infty )} \right. \nonumber \\&\left. \quad - \sum _{k=0}^l \frac{(-1)^k}{k!} \int _{(s,\infty )^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \le \sum _{k=l+1}^\infty \frac{c_2^k k^{k/2}}{k!} \le \frac{\epsilon }{6}.\nonumber \\ \end{aligned}$$
(C.21)

By (ii), we have

$$\begin{aligned}&\left| \left( \int _{(s,\infty )^k}-\int _{(s,L]^k}\right) \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \nonumber \\&\quad \le \int _{\begin{array}{c} (s,\infty )^k \\ \text {some}~ \xi _j>L \end{array}} \left| \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \right| \prod _{i=1}^k d\xi _i\nonumber \\&\quad \le \sum _{j=1}^k \int _{\begin{array}{c} \xi _j>L\\ \xi _l>s, l\ne j \end{array}} c_1^k k^{k/2} \prod _{i=1}^k e^{-\max (0,\xi _i)} \prod _{i=1}^k d\xi _i\nonumber \\&\quad = c_1^k k^{k/2+1} \int _L^{\infty } e^{-\max (0,\xi _j)}d\xi _j \left( \int _s^{\infty } e^{-\max (0,\xi )} d\xi \right) ^{k-1}, \end{aligned}$$
(C.22)

where in the second inequality we used (C.18). Hence, for L large enough,

$$\begin{aligned}&\left| \sum _{k=0}^l \frac{(-1)^k}{k!} \left( \int _{(s,\infty )^k}-\int _{(s,L]^k}\right) \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \nonumber \\&\quad \le \sum _{k=0}^{\infty } \frac{c_1^k k^{k/2+1}}{k!} \int _L^{\infty } e^{-\max (0,\xi _j)}d\xi _j \le c_2 \int _L^{\infty } e^{-\max (0,\xi _j)}d\xi _j \le \frac{\epsilon }{6}, \end{aligned}$$
(C.23)

uniformly in N. Combining (C.21) and (C.23), we have

$$\begin{aligned} \left| \det (1-K_N)_{L^2(s,\infty )} - \sum _{k=0}^l \frac{(-1)^k}{k!} \int _{(s,L]^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \le \frac{\epsilon }{3}. \end{aligned}$$
(C.24)

By the uniform convergence (i), for N large enough,

$$\begin{aligned}&\left| \sum _{k=0}^l \frac{(-1)^k}{k!} \int _{(s,L]^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right. \nonumber \\&\quad \left. - \sum _{k=0}^l \frac{(-1)^k}{k!} \int _{(s,L]^k} \det (K(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \le \frac{\epsilon }{3}. \end{aligned}$$
(C.25)

By the same argument as to get (C.24), we have

$$\begin{aligned} \left| \det (1-K)_{L^2{(s,\infty )} } - \sum _{k=0}^l \frac{(-1)^k}{k!} \int _{(-\infty ,L]^k} \det (K(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \le \frac{\epsilon }{3}. \end{aligned}$$
(C.26)

Combining (C.24), (C.25), (C.26), we have

$$\begin{aligned} {|}\det (1-K_N)_{L^2(s,\infty )} - \det (1-K)_{L^2{(s,\infty )}} | \le \epsilon . \end{aligned}$$
(C.27)

This completes the proof for the \(f_N=1_{(s,\infty )},\forall N\in \mathbb {N}\) case.

Next we consider the general \(f_N\) case. It is enough to show

$$\begin{aligned}&\left| \int _{\mathbb {R}^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k f_N(\xi _i) d\xi _i \right. \nonumber \\&\quad \left. - \int _{(s,\infty )^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \le c^k k^{k/2+1}\epsilon \end{aligned}$$
(C.28)

for large enough N. Indeed, taking the sum over k and using the triangular inequality on the Fredholm explansion, the bound (C.28) gives

$$\begin{aligned} {|} \det (1-f_N K_N)_{L^2(\mathbb {R})} - \det (1-K_N)_{L^2{(s,\infty )}} | \le \sum _{k=1}^{\infty } c^k k^{k/2+1}\epsilon \le C\epsilon . \end{aligned}$$
(C.29)

Then taking \(N\rightarrow \infty \) limit, we find

$$\begin{aligned} \lim _{N\rightarrow \infty } \det (1-f_N K_N)_{L^2(\mathbb {R})} = \lim _{N\rightarrow \infty } \det (1-K_N)_{L^2{(s,\infty )}}. \end{aligned}$$
(C.30)

This completes the proof for general \(f_N\) case. Coming back to the proof of (C.28), in the second term on lhs, the \(\infty \) as the upper limit of the integrals can be replaced by a large L with an error of the form \(c_1^k k^{k/2+1}\epsilon \) as in (C.22). Similarly, in the first term on lhs of (C.28), the \(\infty \) as the upper limit of the integrals can be replaced by L with an error of the form \(c_2^k k^{k/2+1}\epsilon \) and the \(-\infty \) as the lower limit by \(-L\) with an error of the form \(c_3^k k^{k/2+1}\epsilon \) due to (iv). Combining these, we can replace the integrals in (C.28) within \((-L,L]\) for large L with an error \(c_4^k k^{k/2+1}\epsilon \). By a similar argument as to get (C.22), we have

$$\begin{aligned}&\left| \int _{(-L,L]^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k f_N(\xi _i) d\xi _i - \int _{(s,L]^k} \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \prod _{i=1}^k d\xi _i \right| \nonumber \\&\quad \le \sum _{j=1}^k \int _{(-L,s]}d\xi _j \int _{(-L,L]^{k-1}} \prod _{i=1,i\ne j}^k d\xi _i \prod _{i=1}^k |f_N(\xi _i)| \left| \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \right| \nonumber \\&\qquad + \int _{(s,L]^k} \left| \prod _{i=1}^k f_N(\xi _i)-1 \right| \left| \det (K_N(\xi _i,\xi _j))_{1\le i,j\le k} \right| \prod _{i=1}^k d\xi _i . \end{aligned}$$
(C.31)

Using (C.18) and (iii), one observes that the integral over \(\xi _i, i\ne j\) in the first sum is finite. By using (iii), we see that the integral over \(\xi _j\) in the first sum is bounded for large N by \( c_5^k k^{k/2+1} \epsilon \). Similarly for the second term in (C.31), one first sees \(| \prod _{i=1}^k f_N(\xi _i)-1 | \le c_6 \sum _{i=1}^k |f_N(\xi _i)-1|\) since \(f_N\) is bounded. Then using (iii) and (C.18) one sees that it is bounded by \( c_7^k k^{k/2+1} \epsilon \) for large N. Combining all of the above, we get (C.28). \(\square \)

Appendix D: Inverse q-Laplace transforms

In this appendix we discuss inversion formulas of the q-Laplace transform. In Appendix B of [43], we discussed a few inversion formulas for the ordinary Laplace transform, which is summarized as follows. First the Laplace transform is defined as

$$\begin{aligned} \hat{\varphi }(u) = \int _0^{\infty } e^{-ux} \varphi (x) dx, ~ u\in \mathbb {C}. \end{aligned}$$
(D.1)

(In [43] we used the notation \(\tilde{\varphi }\). Here we use \(\hat{\varphi }\) to keep the former for the transformed one, see below.) When \(\varphi (x)\) is a (probability) distribution function on \((0,\infty )\), the Laplace transform is analytic for \(\mathrm{Re}\,u>0\). The usual inversion formula is

$$\begin{aligned} \varphi (x) = \frac{1}{2\pi i} \int _{\delta + i\mathbb {R}} du e^{ux} \hat{\varphi }(u),~x>0, \end{aligned}$$
(D.2)

where \(\delta \) should be taken so that the singularities of \(\hat{\varphi }\) are to the left of the integration contour. If \(\varphi \) is associated with a random variable X, we have \(G(u):=\langle e^{-uX} \rangle = u\hat{\varphi }(u)\). The formula can be restated for the random variable \(Y=\log X\). For instance for the distribution \(F(y)=\mathbb {P}[Y\le y] = \mathbb {P}[X\le e^y] = \varphi (e^y)\), (D.1) is rewritten as

$$\begin{aligned} \hat{\varphi }(u) = \int _{\mathbb {R}} e^{-ue^y+y} F(y) dy =:\tilde{F}(u) . \end{aligned}$$
(D.3)

For discussing the distribution of the O’Connell–Yor polymer model, the following inversion formula was useful.

Proposition D.1

For a random variable Y, set \(G(u) = \langle e^{-ue^Y}\rangle \). The distribution function of Y, which decays as \(F(y) = e^{a y}, a>0\) as \(y\rightarrow -\infty \), is recovered from G(u) as

$$\begin{aligned} F(y) = \frac{1}{2\pi i} \int _{\delta +i\mathbb {R}} d\xi \frac{e^{y \xi }}{\Gamma (\xi +1)} \int _0^{\infty } u^{\xi -1} G(u)du, \end{aligned}$$
(D.4)

where \(0<\delta <a\). The corresponding density function \(f(y)=F'(y)\), if it exists, is given by

$$\begin{aligned} f(y) = \frac{1}{2\pi i} \int _{\delta +i\mathbb {R}} d\xi \frac{e^{y \xi }}{\Gamma (\xi )} \int _0^{\infty } u^{\xi -1} G(u) du. \end{aligned}$$
(D.5)

The formulas discussed in this appendix are q-analogues of the above.

Suppose we have a function \(f(n),n\in \mathbb {Z}\) and denote by \(\tilde{f}_q(z)\) its q-Laplace transform,

$$\begin{aligned} \tilde{f}_q(z) := \sum _{n\in \mathbb {Z}} \frac{f(n)}{(zq^n;q)_{\infty }}, \end{aligned}$$
(D.6)

In this appendix, we mainly consider the case where \(f(n),n\in \mathbb {Z}\) is a discrete probability density function, i.e., \(f(n)\ge 0, \sum _{n\in \mathbb {Z}} f(n)=1\), for which the q-Laplace transform converges and analytic except \(z\ne q^n, n\in \mathbb {Z}\). By using the fact that the q-exponential function tends to the usual exponential function in a \(q\rightarrow 1\) limit [see the comment after (A.5)], one sees that this formula goes to (D.1) as

$$\begin{aligned} \tilde{f}_q(-(1-q)u) = \sum _{n\in \mathbb {Z}} \frac{f(n)}{(-(1-q)q^nu;q)_\infty } \rightarrow \int _\mathbb {R}dy e^{y-ue^y} f(y) = \hat{f}(u), \end{aligned}$$
(D.7)

where in the limit (\(\rightarrow \)) we set \(n=y/\log q\) and took \(q\rightarrow 1\). An inverse formula for the q-Laplace transform is given by

$$\begin{aligned} f(n) = \int _\gamma \frac{dz}{2\pi i} q^n(q^{n+1}z;q)_\infty \tilde{f}_q(z), \end{aligned}$$
(D.8)

where \(\gamma \) is a osing \(\mathbb {R}_+\) clockwise, see Fig. 6. Because of the factor \((q^{n+1}z;q)_\infty \), the poles at \(z=q^{-k},k=n+1,n+2,\ldots \) vanish and the contour can be taken to be around the poles at \(z=1,2,\ldots , n\), but the infinite contour in Fig. 6 has an advantage that it can be used for any n.

Fig. 6
figure 6

The contour \(\gamma \)

In the \(q\rightarrow 1\) limit, with \(n=y/\log q\) and the change of variable \(z=-(1-q)u\), tends to (D.2) as

$$\begin{aligned} f(n) = q^n(1-q)\int _{\gamma }\frac{du}{2\pi i} (-(1-q)q^{n+1}u;q)_\infty \tilde{f}_q(-(1-q)u) \rightarrow \int _{\delta +i\mathbb {R}} \frac{du}{2\pi i}e^{u e^y}\hat{f}(u). \end{aligned}$$
(D.9)

There is also an inversion formula corresponding to (D.5).

Proposition D.2

When \(f(n),n\in \mathbb {Z}\) is a discrete probability density function,

$$\begin{aligned} f(n) = -\int _{C_0} \frac{dx}{2\pi i x^{n+1}} \frac{(q;q)_{\infty }}{(qx;q)_{\infty }} \sum _{k\in \mathbb {Z}} (qx)^k R_k(\tilde{f}_q). \end{aligned}$$
(D.10)

Here \(R_k(\tilde{f}_q)\) is the residue of \(\tilde{f}_q(z)\) at the pole \(z=q^{-k}\) and \(C_0\) is a small contour around the origin.

Proof

It is equivalent to showing

$$\begin{aligned} \sum _{n\in \mathbb {Z}} f(n) x^n = -\frac{(q;q)_{\infty }}{(qx;q)_{\infty }} \sum _{k\in \mathbb {Z}} (qx)^k R_k(\tilde{f}_q), \end{aligned}$$
(D.11)

for \(x=e^{i\theta },\theta \in (-\pi ,\pi ]\). In fact considering the contour integral around the origin of both sides of (D.11) divided by \(x^{n+1}\), we get (D.10). By (D.8), this is further equivalent to showing

$$\begin{aligned} -\sum _{n\in \mathbb {Z}}x^n \int _{\gamma } \frac{dz}{2\pi i} \tilde{f}_q(z) q^n(q^{n+1}z;q)_{\infty } = \frac{(q;q)_{\infty }}{(qx;q)_{\infty }} \sum _{k\in \mathbb {Z}} (qx)^k R_k(\tilde{f}_q). \end{aligned}$$
(D.12)

Taking the poles at \(z=q^{-k},k\in \mathbb {Z}\), one can rewrite lhs of (D.12) as

$$\begin{aligned} \sum _{k\in \mathbb {Z}} R_k(\tilde{f}_q) \sum _{n\in \mathbb {Z}} (qx)^n(q^{n-k+1};q)_{\infty }&= \sum _{k\in \mathbb {Z}} (qx)^k R_k(\tilde{f}_q) \sum _{n\in \mathbb {Z}} (qx)^{n-k}(q^{n-k+1};q)_{\infty }. \end{aligned}$$
(D.13)

The last sum in (D.13) can be taken owing to the q-binomial theorem (A.2),

$$\begin{aligned} \sum _{n\in \mathbb {Z}} (qx)^{n-k}(q^{n-k+1};q)_{\infty }&= \sum _{n=k}^{\infty } (qx)^{n-k}(q^{n-k+1};q)_{\infty } = \frac{(q;q)_{\infty }}{(qx;q)_{\infty }}. \end{aligned}$$
(D.14)

Substituting this into (D.13), we get (D.12). \(\square \)

The last sum in (D.10) formally seems to be a consequence of taking poles at \(z=q^{-k},k\in \mathbb {Z}\) of the integral, \(\int _{\gamma } \frac{dz}{2\pi i z} z^{\xi } \tilde{f}_q(z)\) (though this is not true since \(z^{\xi }\) has a cut along \(\mathbb {R}_+\)). Setting \(x=q^{-\xi }\) and taking the \(q\rightarrow 1\) limit of this and recalling the factor \((q;q)_\infty /(qx;q)_\infty \) can be written in terms of the q-Gamma function (A.6) which tends to the Gamma function as \(q\rightarrow 1\), one would observe that (D.10) may be regarded as a q-analogue of (D.5).

As a corollary of proposition D.2, we get a formula for the distribution function \(F(n)=\mathbb {P}[X\le n], n\in \mathbb {N}\). It reads

$$\begin{aligned} F(n) = -\int _{C_0} \frac{dx}{2\pi i x^{n+1}} \frac{(q;q)_{\infty }}{(x;q)_{\infty }} \sum _{k\in \mathbb {Z}} (qx)^k R_k(\tilde{f}_q). \end{aligned}$$
(D.15)

For the density function, \(f(n):=F(n)-F(n-1)\), this gives (D.10). (D.15) is an analogue of (D.4). If we introduce the generating function \(Q(x)=\sum _{n\in \mathbb {Z}} F(n) x^n\), clearly

$$\begin{aligned} F(n) = \int _{C_0} \frac{dx}{2\pi i x^{n+1}} Q(x) \end{aligned}$$
(D.16)

and (D.15) is equivalent to

$$\begin{aligned} Q(x) = \frac{(q;q)_{\infty }}{(x;q)_{\infty }} \sum _{k\in \mathbb {Z}} (q x)^k R_k(\tilde{f}_q). \end{aligned}$$
(D.17)

Suppose further that a random variable X having the density \(f(n),n\in \mathbb {Z}\) is written as \(X=X_0+\chi \) where \(X_0\) and \(\chi \) are independent random variables on \(\mathbb {Z}\). By the independence, the three quantities, Q(x), \(Q_0(x)=\sum _{n\in \mathbb {Z}} \mathbb {P}[X_0 \le n] x^n, g(x)=\sum _{n\in \mathbb {Z}} \mathbb {P}[\chi = n] x^n\), are related as \(Q(x) = g(x)Q_0(x)\). Combining this and (D.16), (D.17), we find a formula for the distribution of \(X_0\),

$$\begin{aligned} \mathbb {P}[X_0\le n]&= \int _{C_0} \frac{dx}{2\pi i x^{n+1}} \frac{(q;q)_{\infty }}{(x;q)_{\infty }g(x)} \sum _{k\in \mathbb {Z}} (qx)^k R_k(\tilde{f}_q). \end{aligned}$$
(D.18)

Noting \(\tilde{f}_q(\zeta )= \langle \frac{1}{(\zeta q^X;q)}\rangle \), this is a formula which gives the distribution function of \(X_0\) in terms of the q-Laplace transform of X.

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Imamura, T., Sasamoto, T. Fluctuations for stationary q-TASEP. Probab. Theory Relat. Fields 174, 647–730 (2019). https://doi.org/10.1007/s00440-018-0868-3

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Keywords

  • Asymmetric simple exclusion process
  • Kardar–Parisi–Zhang universality
  • Integrable probability
  • Macdonald processes
  • Frobenius determinant

Mathematics Subject Classification

  • 82C22
  • 60K35
  • 33D52