Stationary stochastic Higher Spin Six Vertex Model and q-Whittaker measure

Abstract

In this paper we consider the Higher Spin Six Vertex Model on the lattice ${\mathbb {Z}}_{\ge 2} \times {\mathbb {Z}}_{\ge 1}$. We first identify a family of translation invariant measures and subsequently we study the one point distribution of the height function for the model with certain random boundary conditions. Exact formulas we obtain prove to be useful in order to establish the asymptotic of the height distribution in the long space-time limit for the stationary Higher Spin Six Vertex Model. In particular, along the characteristic line we recover Baik–Rains fluctuations with size of characteristic exponent 1/3. We also consider some of the main degenerations of the Higher Spin Six Vertex Model and we adapt our analysis to the relevant cases of the q-Hahn particle process and of the Exponential Jump Model.

Appendices

Preliminaries on q-deformed quantities

Along the course of the paper we largely made use of q-deformed quantities, such as q-Pochhammer symbols and q-hypergeometric series. The reader might consider these as fairly common and established notions, but, for the sake of completeness, we still like to dedicate this appendix to recall their definitions.

Assuming q is a parameter in the interval [0, 1), we define the q-Pochhammer symbol

$$\begin{aligned} (z;q)_n = {\left\{ \begin{array}{ll} (1-z)(1-zq) \cdots (1-zq^{n-1}), \quad \quad &{}\text {if } n \in {\mathbb {Z}}_{>0},\\ 1, &{}\text {if } n=0,\\ (1-zq^{n})^{-1} (1-zq^{n-1})^{-1} \cdots (1-zq)^{-1}, &{} \text {if } n \in {\mathbb {Z}}_{<0}, \end{array}\right. } \end{aligned}$$

(A.1)

for every meaningful $z \in {\mathbb {C}}$. We also denote the product of multiple q-Pochhammer symbols of the same order in the compact notation

$$\begin{aligned} (z_1;q)_n \cdots (z_k;q)_n = (z_1,\dots ,z_k;q)_n. \end{aligned}$$

(A.2)

When n is positive, the q-Pochhammer symbol (A.1) is a polynomial in z and it admits the expansion

$$\begin{aligned} (z;q)_n=\sum _{k=0}^n (-z)^k q^{\left( {\begin{array}{c}k\\ 2\end{array}}\right) } \left( {\begin{array}{c}n\\ k\end{array}}\right) _{q}, \end{aligned}$$

(A.3)

where we introduced the q-binomial

$$\begin{aligned} \left( {\begin{array}{c}n\\ k\end{array}}\right) _{q} = \frac{(q;q)_n}{(q;q)_k (q;q)_{n-k}}. \end{aligned}$$

(A.4)

The the q-binomial admits the combinatorial expansion

$$\begin{aligned} \left( {\begin{array}{c}n\\ k\end{array}}\right) _{q}= \sum _{\begin{array}{c} I \subset \{1,\dots ,n\} \\ |I|=k \end{array}} q^{\Vert I \Vert - \left( {\begin{array}{c}k+1\\ 2\end{array}}\right) }, \end{aligned}$$

(A.5)

with $I=\{i_1,\dots , i_k\}$ and $\Vert I \Vert = i_1+ \cdots +i_k$.

When we let the integer n grow to $+ \infty$, we see that the product in the left hand side of (A.1) is convergent and hence we can define

$$\begin{aligned} (z;q)_{\infty } = \prod _{j \ge 0} (1- z q^{j}). \end{aligned}$$

(A.6)

An important result concerning q-Pochhammer symbols is the summation identity

$$\begin{aligned} \sum _{k \ge 0} \frac{(a;q)_k}{(q;q)_k}z^k = \frac{(za;q)_\infty }{(z;q)_\infty } \qquad \text {for } a\in {\mathbb {C}}, |z|<1, \end{aligned}$$

(A.7)

which can be found in [41], Theorem 12.2.5. and it is usually called q-binomial theorem. A slightly more general version of summation (A.7) is the so called q-Gauss summation ([41], Theorem 12.2.4)

$$\begin{aligned} \sum _{n \ge 0} \left( \frac{c}{a b} \right) ^n \frac{(a,b;q)_n}{(c,q;q)_n} = \frac{(c/a, c/b;q)_\infty }{(c, c/(ab);q)_\infty }, \quad \text {for } |c/(ab)|<1, \quad \text {or }b\in q^{{\mathbb {Z}}_{<0}} .\nonumber \\ \end{aligned}$$

(A.8)

The q-hypergeometric series

$$\begin{aligned} {}_{r+1} \phi _{r} \left( \begin{array}{llllll} &{}a_1,&{} a_2, &{} \dots , &{} a_{r+1} &{}\\ &{}b_1,&{}b_2, &{}\cdots ,&{} b_{r}&{} \end{array} \Big | q, z \right) = \sum _{k \ge 0} \frac{ (a_1 , \dots , a_{r+1};q)_k }{ (b_1, \dots , b_{r},q;q)_k }z^k, \end{aligned}$$

(A.9)

is defined for generic parameters $a_1, \dots , a_{r+1} \in {\mathbb {C}}$, $b_1, \dots , b_{r} \in {\mathbb {C}} \setminus q^{{\mathbb {Z}}_{<0}}$ and $|z|<1$. In the case when at least one of the $a_j$ is of the form $q^{-k}$, for some non-negative integer k, the q-hypergeometric series (A.9) becomes a finite sum and its definition holds also for more general complex numbers z. The regularized terminating q-hypergeometric function is also defined as

$$\begin{aligned} {}_{r+1} {\bar{\phi }}_{r} \left( \begin{array}{llllll} &{}q^{-n},&{} a_1, &{} \dots , &{} a_r &{}\\ &{}b_1,&{}b_2, &{}\cdots ,&{} b_r &{} \end{array} \Big | q, z \right) = \sum _{k = 0}^n z^k \frac{(q^{-n};q)_k}{(q;q)_k} \prod _{j=1}^r (a_j;q)_k (q^k b_j ;q)_{n-k}.\nonumber \\ \end{aligned}$$

(A.10)

In Sect. 4 we used the q-analog of the Chu–Vandermonde identity ( [41], (12.2.17)) that we report as

$$\begin{aligned} {}_{2} {\bar{\phi }}_{1} \left( \begin{array}{l} q^{-n}, a\\ c \end{array} \Big | q, q \right) = \frac{(c/a;q)_n}{(c;q)_n}a^n. \end{aligned}$$

(A.11)

In the paper we also made use of functions , defined as

(A.12)

They are related to the more classical q-polygamma function [69]

$$\begin{aligned} \varvec{\psi }_q(\theta ) = - \log (1-q) + \log (q) \sum _{n \ge 0} \frac{q^{n+\theta }}{1- q^{n + \theta }}, \end{aligned}$$

since

(A.13)

and

(A.14)

The inverse of the infinite q-Pochhammer symbol (A.6) is often called q-exponential and through it one can define a q-deformed notion of the common Laplace transform. For a given $f \in \ell ^1({\mathbb {Z}})$ the function

$$\begin{aligned} {\widetilde{f}}(\zeta ) = \sum _{n \in {\mathbb {Z}}} \frac{f(n)}{(q^n \zeta ;q)_\infty } \qquad \text {for }\zeta \in {\mathbb {C}} \setminus q^{{\mathbb {Z}}} \end{aligned}$$

(A.15)

is the q-Laplace transform of f. As for the usual Laplace transform, the operation $f \mapsto {\widetilde{f}}$ admits an inverse. This is discussed, for example, in [39] and we do not report the exact form of the inverse q-Laplace transform as we do not explicitly make use of it during this paper.

Bounds for $\phi _l, \psi _l, \Phi _x, \Psi _x$

We collect here some useful bounds for the quantities $\phi _l, \psi _l, \Phi _x, \Psi _x$ defined in eqs. (5.4) to (5.7). Terms $\Phi _x,\Psi _x$ can be further decomposed as

$$\begin{aligned} \Phi _x(n)= & {} \Phi _x^{(1)}(n) + \Phi _x^{(2)}(n),\\ \Psi _x(n)= & {} \Psi _x^{(1)}(n) + \Psi _x^{(2)}(n), \end{aligned}$$

obtained separating from the integration (5.6) (resp. (5.7)) the contribution of pole (resp. $z= v$) from that of other poles. Their exact expression was given in eqs. (5.16) to (5.19).

Proposition B.1

Let . Then, for all fixed x, there exist constants $\Gamma _1, \Gamma _2, \Gamma _3, \Gamma _4>0$, such that

$$\begin{aligned} |\phi _l(n)|,|\psi _l(n)|,|\Phi _x(n)|,|\Phi ^{(2)}_x(n)| < \Gamma _1 e^{-\Gamma _2 |n|} \qquad \text {for all }n \in {\mathbb {Z}} \end{aligned}$$

(B.1)

and

$$\begin{aligned} |\Psi _x(n)|< {\left\{ \begin{array}{ll} \Gamma _1 e^{-\Gamma _2 |n|} \qquad \text {if }n \in {\mathbb {Z}}_{<0}\\ \Gamma _3 e^{-\Gamma _4 |n|} \qquad \text {if }n \in {\mathbb {Z}}_{\ge 0}. \end{array}\right. } \end{aligned}$$

(B.2)

Moreover $\Gamma _1,\Gamma _2$ can be chosen so that their relative bounds also hold for in the region (1.18) (in this case the parameter b appearing in the definition of $\tau$ (5.8) satisfies ).

Proof

We start with the terms $\phi _l,\Phi _x(n),\Phi _x^{(2)}$. Evaluating the complex integrals as sums of residues it is straightforward to get the inequalities

for some constants $\Gamma _1, \Gamma _2$ depending on the integrand functions but not on n.

To obtain a similar bound for the term $\psi _l(n)$ we distinguish two cases. When n is positive we take the contour C to be a circle of radius $r_+$ so that $qv< r_+ < b$. On the other hand, when n is negative we take C to be a circle of radius $r_-$ strictly bigger than c, not containing any of the numbers $\xi _i/s_i$ (we remark that the definition itself of $\tau (n)$ and of numbers b, c is tailor-made for these conditions to be possible). With this choices we easily get

$$\begin{aligned} |\psi _l(n)| \le \text {const.}\frac{1}{\tau (n)} \left( \mathbbm {1}_{n\ge 0} r_+^{|n|} + \mathbbm {1}_{n<0} \frac{1}{r_-^{|n|}} \right) \le \Gamma _1 e^{-\Gamma _2 |n|}. \end{aligned}$$

An argument equivalent to that used for $\psi _l$ can be carried to show (B.2). The only difference here is that the radius $r_+$ has to be chosen so that $v< r_+ < b$ and hence we cannot extend this bound to the region $qv <b \le v$. $\square$

Proposition B.2

Let v satisfy (1.11) and or possibly (1.18). Then, for each x, there exist constants $\Gamma _1, \Gamma _2>0$ such that

$$\begin{aligned} |\phi _l (n) \Psi _x (n) |,|\Phi _x^{(2)}(n) \Psi _x (n) | < \Gamma _1 e^{-\Gamma _2 |n|}. \end{aligned}$$

(B.3)

Proof

From Proposition B.1 we see that we only have to prove (B.3) for positive n’s. When this is the case we see directly from the integral expression (5.7) and (5.4) that we can bound both $|\phi _l(n) \Psi _x(n)|$ and $|\Phi ^{(2)}_x(n) \Psi _x(n)|$ with some quantity proportional to

$$\begin{aligned} \frac{|v + \epsilon |^n}{\min _{k \ge 2} |\xi _k s_k|^n }, \end{aligned}$$

(B.4)

by simply taking the C contour as a circle of radius $v + \epsilon$, for $\epsilon$ being sufficiently small. Due to the condition

$$\begin{aligned} v < \min _{k \ge 2}|\xi _k s_k|, \end{aligned}$$

we see that $\epsilon$ can be chosen so that (B.4) decays to zero and this completes the proof. $\square$

Proposition B.3

Let satisfy (1.18). Then, for each fixed x, there exist constants $\Gamma _1, \Gamma _2>0$ such that

$$\begin{aligned} \Big |f(n) \Phi _x^{(1)}(n) \Psi _x^{(2)} (n) \Big | < \Gamma _1 e^{-\Gamma _2 |n|}. \end{aligned}$$

Proof

We use the integral expression (5.19). When n is positive we take the integration contour $C_1$ to be a circle of radius $qv + \epsilon$. A bound we can easily obtain is

On the other hand, when n is negative we chose the contour $C_1$ as a circle of radius $v - \epsilon$ to get a bound like

In both cases condition (1.18) allows us to select $\epsilon$ small enough to guarantee exponential decay in |n|. $\square$

Construction of contours

Here we discuss the construction of the steep descent contour C and that of the steep ascent contour D which were used in the asymptotic analysis of the Stationary Higher Spin Six Vertex Model in Sect. 6.

Proposition C.1

Consider fixed real numbers

$$\begin{aligned} 0<v<\varsigma , \qquad 0<q<1 \end{aligned}$$

and assume that

$$\begin{aligned} \varsigma< \inf _{k\ge 2}\{ \xi _k s_k \} \le \sup _{k\ge 2}\{ \xi _k s_k \}< \infty \quad \text {and} \quad 0 \le s_k^2 <1, \qquad \text {for all } k\ge 2. \end{aligned}$$

Take also a number $\rho < \varsigma$ and define the contour

$$\begin{aligned} C_{\rho } = \{ \rho e^{i \vartheta } | \vartheta \in [0,2 \pi ) \}. \end{aligned}$$

Then, for $\rho$ sufficiently close to $\varsigma$ we have

$$\begin{aligned} \frac{d}{d\vartheta } \mathfrak {Re}\{g(\rho e^{i \vartheta })\}< 0 \quad \text {for } 0<\vartheta <\pi , \end{aligned}$$

(C.1)

where g is given in (6.42).

Remark C.2

The result of Proposition C.1 implies that $C_\rho$ is a steep descent contour for $\mathfrak {Re}(g)$ and in particular

• 1. $\max _{z \in C_\rho } \mathfrak {Re}\{g(z)\} = g(\rho )$;

• 2. $\max _{z \in C_\rho } |z| = \rho$.

This easily follows from (C.1) and from the fact that $g({\overline{z}}) = \overline{g(z)}$, which implies that $\mathfrak {Re}(g)$ is symmetric with respect to the real axis.

Proof

Evaluating the derivative we have

$$\begin{aligned} \begin{aligned}&\frac{d}{d\vartheta } \mathfrak {Re}\{g(\rho e^{i \vartheta })\}\\&\quad = \sin \vartheta \, \kappa \left( \sum _{i=0}^{J-1} \frac{ q^i u \rho }{1 + q^{2i} u^2 \rho ^2 - 2 q^i u \rho \cos \vartheta } \right) \\&\qquad + \sin \vartheta \frac{1}{x} \sum _{k=2}^x \sum _{j\ge 0} \left( \frac{\frac{q^j \rho }{\xi _k s_k} }{ 1 + \left( \frac{q^j \rho }{\xi _k s_k} \right) ^2 -2\frac{q^j \rho }{\xi _k s_k} \cos \vartheta } - \frac{\frac{q^j s_k^2 \rho }{\xi _k s_k} }{ 1 + \left( \frac{q^j s_k^2 \rho }{\xi _k s_k} \right) ^2 -2\frac{q^j s_k^2 \rho }{\xi _k s_k} \cos \vartheta } \right) . \end{aligned}\nonumber \\ \end{aligned}$$

(C.2)

Each term

$$\begin{aligned} \frac{ q^i u \rho }{1 + q^{2i} u^2 \rho ^2 - 2 q^i u \rho \cos \vartheta }, \end{aligned}$$

has a maximum in $\vartheta = 0$ due to the fact that u and $\rho$ have opposite sign, and so does each single one of the summands in the double summation in (C.2), since the generic function

$$\begin{aligned} \frac{ a }{ 1 + a^2 -2 a \cos \vartheta } - \frac{ a \sigma }{ 1 + a^2 \sigma ^2 -2 a \sigma \cos \vartheta } \end{aligned}$$

is decreasing in $0<\vartheta < \pi$, provided that $0<a, \sigma <1$. Now, if $\rho$ is taken sufficiently close to the critical point $\varsigma$, in a neighborhood of $\vartheta =0$, the derivative of $\mathfrak {Re}\{g(\rho e^{i \vartheta })\}$ is negative by construction and, thanks to considerations we just made, it stays negative along the whole half circle. $\square$

The construction of an explicit steepest descent contour D for a general choice of parameters $q, \Xi , {\mathbf{S}}$ becomes more complicated. Therefore we use the next Proposition both to exhibit a contour in a rather simple setting and to implicitly deduce conditions on $q, \Xi , {\mathbf{S}}$ under which our arguments of Sect. 6 are perfectly well posed.

Proposition C.3

For each choice of

$$\begin{aligned} 0<\varsigma<a, \qquad u<0, \qquad 0<\sigma <1, \end{aligned}$$

there exist constants $R_a, R_\sigma , R_q>0$, such that for each choice of parameters $\{ \xi _k \}_{k\ge 2}, \{ s_k \}_{k \ge 2}$, q satisfying

$$\begin{aligned} \varsigma<\inf _{k\ge 2}\{\xi _k s_k\}, \qquad |\xi _k s_k - a|<R_a, \qquad |s_k^2 - \sigma |<R_\sigma , \qquad q<R_q, \end{aligned}$$

we are able to construct a complex contour D encircling the set $\{\xi _k s_k\}_{k \ge 2}$, for which

• 1. $\min _{z \in D} \mathfrak {Re}\{g(z)\} = g(\varsigma )$;

• 2. $\min _{z \in D} |z| = \varsigma$,

where g is given in (6.42).

Proof

To show this result we essentially make use of a continuity argument. We start studying the case when

$$\begin{aligned} q=0, \qquad \xi _k, s_k=a \qquad s_k^2=\sigma \qquad \text {for all } k \ge 2. \end{aligned}$$

With this choice of parameters the function g becomes

$$\begin{aligned} g(z)=- \eta \log (z)+ \kappa \log ( 1- u z) + \log \left( \frac{a-z}{a- \sigma z} \right) + {\mathcal {O}}(x^{-1}), \end{aligned}$$

(C.3)

where we can neglect the contribution of the ${\mathcal {O}}(x^{-1})$ term as we are interested in this result only in the limiting case of $x \rightarrow \infty$. We define the contour D to be the level curve

$$\begin{aligned} D=\left\{ z:\ \mathfrak {Re}\left\{ \log \left( \frac{a-z}{a- \sigma z} \right) \right\} = \log \left( \frac{a-\varsigma }{a- \sigma \varsigma } \right) \right\} , \end{aligned}$$

which is a circle and admit the parametrization

$$\begin{aligned} \left\{ \varsigma + \rho + \rho e^{i \vartheta } |\ \vartheta \in [0, 2 \pi ) \right\} , \end{aligned}$$

with the radius $\rho$ being

$$\begin{aligned} \rho = \frac{a^2 - a\varsigma - a \varsigma \sigma + \varsigma ^2 \sigma }{a+ a\sigma -2 \varsigma \sigma }. \end{aligned}$$

We also report that the leftmost and rightmost extremes of the contour D are respectively $\varsigma$ and $\varsigma + 2 \rho$ and one can easily find that the latter satisfies the inequality

$$\begin{aligned} \varsigma + 2 \rho \le \frac{2 a}{1 + \sigma }. \end{aligned}$$

(C.4)

Along the curve D we are able to calculate

$$\begin{aligned} \frac{d}{d \vartheta } \mathfrak {Re} \left\{ g(\varsigma + \rho + \rho e^{i \vartheta }) \right\} \end{aligned}$$

and to analytically show that its only critical points are $\theta \in {\mathbb {Z}} \pi$. More specifically, substituting in (C.3) the correct expressions of coefficients $\eta , \kappa$ given in (1.21)

$$\begin{aligned} \eta= & {} \frac{a \varsigma ^2 (1- \sigma ) (a- a^2 u + a \sigma -2 \varsigma \sigma + \varsigma ^2 u \sigma ) }{(a- \varsigma )^2 (a- \varsigma \sigma )^2}, \end{aligned}$$

(C.5)

$$\begin{aligned} \kappa= & {} - \frac{a(1-\varsigma u)^2 (1-\sigma )(a^2 - \varsigma ^2 \sigma )}{u (a- \varsigma )^2 (a-\varsigma \sigma )^2}, \end{aligned}$$

(C.6)

we get

$$\begin{aligned} \frac{d}{d \vartheta } \mathfrak {Re} \left\{ g(\varsigma + \rho + \rho e^{i \vartheta }) \right\} = \sin \vartheta ( 1+ \cos \vartheta ) \frac{1}{P(\cos \vartheta )}. \end{aligned}$$

In the last expression P is a polynomial of degree two in the argument and we see that zeros are only achieved on the real axis for $\theta =k\pi$ for $k \in {\mathbb {Z}}$. We can at this point readily verify that, along D the real part of g assumes a minimum at $z=\varsigma$ and a maximum at $z=\varsigma + 2 \rho$ as the function

$$\begin{aligned} -\eta \log (y) + \kappa \log (1-u y) \end{aligned}$$

is increasing for $y>\varsigma$, and one can check this by direct inspection of its first derivative, by making use of expressions (1.21) for $\eta$ and $\kappa$.

We can now use the fact that g is continuous in the parameters $\Xi , {\mathbf{S}}, q$ for z belonging to D and the fact that, by construction, it will always have a critical point in $z=\varsigma$, to state the existence of neighborhoods respectively of $a, \sigma$ and 0 in which every choice of $\xi _k s_k , s^2_k$ and q will preserve the steepest descent properties 1 and 2. $\square$