Random Walk on Nonnegative Integers in Beta Distributed Random Environment

Abstract

We consider random walks on the nonnegative integers in a space-time dependent random environment. We assume that transition probabilities are given by independent $\textrm{Beta}(\mu ,\mu )$ distributed random variables, with a specific behaviour at the boundary, controlled by an extra parameter $\eta$. We show that this model is exactly solvable and prove a formula for the mixed moments of the random heat kernel. We then provide two formulas that allow us to study the large-scale behaviour. The first involves a Fredholm Pfaffian, which we use to prove a local limit theorem describing how the boundary parameter $\eta$ affects the return probabilities. The second is an explicit series of integrals, and we show that non-rigorous critical point asymptotics suggest that the large deviation behaviour of this half-space random walk in random environment is the same as for the analogous random walk on ${\mathbb {Z}}$.

Appendices

Appendix A: Pfaffians and Fredholm Pfaffians

In this section we provide some background on the notions of Pfaffians and Fredholm Pfaffians, and give conditions for them to be well-defined. The Pfaffian of a skew-symmetric $2k\times 2k$ matrix $A=(a_{i,j})_{i,j=1}^{2k}$ is defined by

$$\begin{aligned} \textrm{Pf}(A) = \frac{1}{2^k k!} \sum _{\sigma \in S_{2k}} \prod _{i=1}^k a_{\sigma (2k-1)}a_{\sigma (2k)}, \end{aligned}$$

(A.1)

and has the property that $\det (A)=\textrm{Pf}(A)^2$. The notion of Fredholm Pfaffian was introduced in [Rai00], as an analogue of the Fredholm determinant. Recall that given a measure space $({\mathbb {X}}, \mu )$ and a kernel $K:{\mathbb {X}}\times {\mathbb {X}}\rightarrow {\mathbb {R}}$, we define the Fredholm determinant $\det (I+K)_{L^2({\mathbb {X}},d\mu )}$ (we will omit the reference measure $d\mu$ when considering the Lebesgue measure on a subset of ${\mathbb {R}}^n$ or when considering the measure $\frac{1}{2{\textbf{i}}\pi }dz$ on a contour of the complex plane, which is the case in this paper) by the series expansion

$$\begin{aligned} \det (I+K)_{L^2({\mathbb {X}},\mu )} = 1+\sum _{k=1}^{+\infty } \frac{1}{k!} \int _{{\mathbb {X}}}d\mu (x_1) \dots \int _{{\mathbb {X}}}d\mu (x_k) \det \left( K(x_i,x_j) \right) _{i,j=1}^k,\nonumber \\ \end{aligned}$$

(A.2)

provided the right-hand-side is a convergent series (see [Sim05] for a more general definition). When proving that such expansions are bounded, one often uses Hadamard’s bound: for a $k\times k$ matrix M such that $\vert m_{i,j}\vert \leqslant a_ib_j$ for all $1\leqslant i,j\leqslant k$,

$$\begin{aligned} \vert \det (M)\vert \leqslant k^{k/2} \prod _{i=1}^ka_ib_i. \end{aligned}$$

(A.3)

Consider now a skew-symmetric, matrix-valued kernel

$$\begin{aligned} K(x,y) = \begin{pmatrix} K_{11}(x,y) &{} K_{12}(x,y) \\ K_{21}(x,y ) &{} K_{22}(x,y) \end{pmatrix},\;\; x,y\in {\mathbb {X}}. \end{aligned}$$

The Fredholm Pfaffian of K, denoted $\textrm{Pf}(J+K)_{L^2({\mathbb {X}}\times \lbrace 1,2\rbrace , d\mu )}$ (again, we will often omit the reference measure $d\mu$), is defined by the series expansion

$$\begin{aligned} \textrm{Pf}(J+K)_{L^2({\mathbb {X}}\times \lbrace 1,2\rbrace , d\mu )} =1+ \sum _{k=1}^{+\infty } \frac{1}{k!} \int _{{\mathbb {X}}}d\mu (x_1) \dots \int _{{\mathbb {X}}}d\mu (x_k) \textrm{Pf}\left( K(x_i,x_j) \right) _{i,j=1}^k, \nonumber \\ \end{aligned}$$

(A.4)

provided the series converges.

Lemma A.1

Let $K:{\mathbb {X}}\times X\rightarrow {\mathbb {R}}$ be a $2\times 2$ matrix valued skew symmetric kernel. Assume that there exist functions $f_{11}, f_{12}, f_{22}\in L^2({\mathbb {X}}, d\mu )$ such that for all $x,y\in {\mathbb {X}}$,

$$\begin{aligned} \vert K_{11}(x,y) \vert \leqslant f_{11}(x), \; \vert K_{12}(x,y)\vert \leqslant f_{12}(x), \; \vert K_{22}(x,y)\vert \leqslant f_{22}(y). \end{aligned}$$

Then, the Fredholm Pfaffian expansion of $\textrm{Pf}(J+K)_{L^2({\mathbb {X}}\times \lbrace 1,2\rbrace , d\mu )}$ in (A.4) is convergent.

Moreover, if a sequence of $2\times 2$ matrix valued skew symmetric kernels $K^{(n)}$ converges pointwise to K as n goes to infinity, and for all $x,y\in {\mathbb {X}}$ we have the bounds (uniformly in n)

$$\begin{aligned} \vert K^{(n)}_{11}(x,y) \vert \leqslant f_{11}(x),\; \vert K^{(n)}_{12}(x,y)\vert \leqslant f_{12}(x),\; \vert K^{(n)}_{22}(x,y)\vert \leqslant f_{22}(y), \end{aligned}$$

then

$$\begin{aligned} \lim _{n\rightarrow +\infty } \textrm{Pf}(J+K^{(n)})_{L^2({\mathbb {X}}\times \lbrace 1,2\rbrace , d\mu )} = \textrm{Pf}(J+K)_{L^2({\mathbb {X}}\times \lbrace 1,2\rbrace , d\mu )}. \end{aligned}$$

(A.5)

Proof

In the series expansion (A.4), we use the fact that

$$\begin{aligned} \left| \textrm{Pf}\left( K(x_i,x_j) \right) _{i,j=1}^k \right| = \sqrt{\left| \det \left( K(x_i,x_j) \right) _{i,j=1}^k \right| }, \end{aligned}$$

which, using Hadamard’s bound (A.3), is bounded by

$$\begin{aligned} (2k)^{k/2} \prod _{i=1}^k f_{11}(x_i)f_{12}(x_i)f_{12}(y_i)f_{22}(y_i). \end{aligned}$$

Using Cauchy-Schwarz inequality, and the fact that $f_{11}, f_{12}, f_{13}\in L^2({\mathbb {X}}, d\mu )$, yields that the series expansion in (A.4) is bounded by

$$\begin{aligned} \sum _{k=0}^{+\infty } \frac{1}{k!} \left( \Vert f_{11}\Vert _2 \Vert f_{12}\Vert _2^2 \Vert f_{22} \Vert _2 \right) ^{k/4} (2k)^{k/2}, \end{aligned}$$

which is finite. The convergence (A.5) then follows from applying the dominated convergence theorem in the integrals and the series. $\quad \square$

Appendix B: Local Limit Theorem for the Full-space Beta RWRE

Let us start by recalling a formula for the moments of ${\textsf{P}}^{{\mathbb {Z}}}(x,0)$, defined in (1.12). For $x_1\geqslant \dots \geqslant x_k$ with $t+x_i$ even, we have

$$\begin{aligned}{} & {} {\mathbb {E}}\left[ \textsf{P}^{{\mathbb {Z}}}_{0,t}(x_1,0)\dots \textsf{P}^{{\mathbb {Z}}}_{0,t}(x_k,0)\right] \nonumber \\{} & {} \quad = (\alpha +\beta )_k \int _{\gamma _1} \frac{dz_1}{2 \pi {\textbf{i}}}\dots \int _{\gamma _k} \frac{dz_k}{2 \pi {\textbf{i}}} \prod _{1 \leqslant a< b \leqslant k} \nonumber \\{} & {} \qquad \times \frac{z_a-z_b}{z_a-z_b-1} \prod _{i=1}^k \left( \frac{(\alpha +z_i)^2}{z_i(z_i+\alpha +\beta )} \right) ^{t/2} \left( \frac{z_i+\alpha +\beta }{z_i} \right) ^{\frac{x_i}{2}+1} \frac{1}{(z_i+\alpha +\beta )^2}, \end{aligned}$$

(B.1)

where the contours $\gamma _1, \dots , \gamma _k$ all contain 0, exclude $-\alpha -\beta$, and are nested in such a way that $\gamma _i$ contains $\gamma _j+1$ for all $1\leqslant i<j\leqslant k$. This formula is a variant of the moment formula from [BC17, Proposition 1.11], given in [TLD16], see also [CG17, (3.1)]. We may deform the contours to become a vertical line $\eta +{\textbf{i}}{\mathbb {R}}$, with $-\alpha -\beta<\eta <0$. Once the contours are all the same, using the symmetrization identity

$$\begin{aligned} \sum _{\sigma \in S_k} \sigma \left( \prod _{a<b} \frac{z_a-z_b-1}{z_a-z_b} \right) = k!, \end{aligned}$$

(B.2)

we obtain that

$$\begin{aligned} {\mathbb {E}}\left[ \textsf{P}^{{\mathbb {Z}}}_{0,t}(x,0)^k\right] = (\alpha +\beta )_k \int _{\eta +{\textbf{i}}{\mathbb {R}}} \frac{dz_1}{2 \pi {\textbf{i}}}\dots \int _{\eta +{\textbf{i}}{\mathbb {R}}} \frac{dz_k}{2 \pi {\textbf{i}}} \det \left( \frac{1}{z_i-z_j-1} \right) _{i,j=1}^k \prod _{i=1}^k f_{t,x}(z_i),\nonumber \\ \end{aligned}$$

(B.3)

where

$$\begin{aligned} f_{t,x}(z) = \left( \frac{(\alpha +z)^2}{z(z+\alpha +\beta )} \right) ^{t/2} \left( \frac{z+\alpha +\beta }{z} \right) ^{\frac{x}{2}+1} \frac{1}{(z+\alpha +\beta )^2}. \end{aligned}$$

We refer to [BC14, Proposition 3.2.2] for similar type of derivations of determinantal formulas. We may now take the generating series to compute the Hankel transform, and obtain

$$\begin{aligned} {\mathbb {E}}\left[ F_{\alpha +\beta }(\zeta \textsf{P}^{{\mathbb {Z}}}_{0,t}(x,0))\right] = \det (I+\zeta L)_{L^2(\eta +{\textbf{i}}{\mathbb {R}})}, \end{aligned}$$

(B.4)

where the kernel L is given by

$$\begin{aligned} L(z,z') = \frac{1}{z-z'-1}f_{t,x}(z,z'). \end{aligned}$$

Let us consider the scalings and changes of variables

$$\begin{aligned} \zeta ={\tilde{\zeta }}\sqrt{t}, \,\; x=\frac{\beta -\alpha }{\beta +\alpha }t+{\tilde{x}}\sqrt{t},\;\; z=\frac{\sqrt{2t \alpha \beta }}{{\tilde{z}}},\;\; z'=\frac{\sqrt{2t \alpha \beta }}{{\tilde{z}}'}. \end{aligned}$$

(B.5)

Under these scalings, we see—the computations are similar as in Sect. 4.4—that we have the pointwise convergence of the kernel

$$\begin{aligned} \sqrt{t}L(z,z') \xrightarrow [t\rightarrow \infty ]{} L^{\infty }({\tilde{z}}, {\tilde{z}}'):= \mathbb {1}_{{\tilde{z}}={\tilde{z}}'} \frac{1}{\sqrt{2\alpha \beta }} \exp \left( \frac{{\tilde{z}}^2}{2}+\frac{x{\tilde{z}}}{\sigma }\right) , \end{aligned}$$

where $\sigma$ is as in Proposition 1.13. Furthermore, the kernel can be bounded in a way similar as in (4.34), that is, there exist constants $c, c'>0$ such that

$$\begin{aligned} \left| L(z,z')\frac{\mathrm d z}{\mathrm d {\tilde{z}}}\right| \leqslant \left| \frac{c}{c'-{\tilde{z}}^2}\right| , \end{aligned}$$

where we recall that $g_{\sigma }$ is the density function of the Gaussian distribution with variance $\sigma ^2= \frac{4\alpha \beta }{(\alpha +\beta )^2}$, as in the statement of Proposition 1.13. Thus, we may apply dominated convergence in the Fredholm determinant expansion. We find that

$$\begin{aligned} \det (I+\zeta L)_{L^2(\eta +{\textbf{i}}{\mathbb {R}})} \xrightarrow [t\rightarrow \infty ]{} \det (I+{\tilde{\zeta }} L^{\infty })_{L^2({\textbf{i}}{\mathbb {R}})} =\exp \left( {\tilde{\zeta }} \int _{{\textbf{i}}{\mathbb {R}}} \frac{dz}{2{\textbf{i}}\pi } L^{\infty }(z,z)\right) = \exp \left( \frac{{\tilde{\zeta }} g_{\sigma }({\tilde{x}}) }{\alpha +\beta }\right) . \end{aligned}$$

Using (B.4) and Proposition 4.5, this shows that under the scalings (B.5)

which concludes the proof of Proposition 1.13.

Appendix C: Critical Point Asymptotics

In this Section, we show that a critical point (non-rigorous) asymptotic analysis of the formula from Proposition 1.9 yields the statement from Conjecture 1.16.

All constants arising in the asymptotic analysis take the simplest possible form when they are written as functions of a real number $\theta >\mu$, which corresponds to the location of the critical point in the asymptotic analysis below. We start with setting $\zeta _t(y)=e^{t a_{\theta } -t^{1/3} b_{\theta } y}$, as in Lemma 4.6, where the constants $a_{\theta }, b_{\theta }$ will be the same as in Conjecture 1.16, and we explain below how these constants were determined. We may restate Proposition 1.9, gathering all factors depending on t, as

$$\begin{aligned}{} & {} {\mathbb {E}}\left[ F_{\mu +\eta } \left( -\zeta _t(y) {\textsf{P}}_{0,2t}(x_{\theta }t,1)/4 \right) \right] \nonumber \\{} & {} \quad = 1+\sum _{\ell =1}^{\infty } \frac{1}{\ell !} \int _{\frac{1}{2}+e^{-{\textbf{i}}\pi /3} \infty }^{\mu +\frac{1}{2} +e^{{\textbf{i}}\pi /3} \infty } \frac{d z_1}{2 \pi {\textbf{i}}} \dots \int _{\mu +\frac{1}{2}+e^{- {\textbf{i}}\pi /3} \infty }^{\mu +\frac{1}{2}+e^{{\textbf{i}}\pi /3} \infty } \frac{d z_{\ell }}{2 \pi {\textbf{i}}} \nonumber \\{} & {} \qquad \oint _{\gamma } \frac{dw_1}{2 \pi {\textbf{i}}} \dots \oint _{\gamma } \frac{d w_{\ell }}{2 \pi {\textbf{i}}} \nonumber \\{} & {} \qquad \prod _{i=1}^{\ell } \frac{\pi }{\sin (\pi (w_i-z_i))} \det \left[ \frac{1}{z_i-w_j} \right] _{i,j=1}^{\ell } \prod _{1 \leqslant i <j \leqslant \ell } \frac{\Gamma (z_i+w_j) \Gamma (w_i+z_j)}{\Gamma (w_i+w_j) \Gamma (z_i+z_j)} \nonumber \\{} & {} \qquad \times \prod _{i=1}^{\ell }\frac{ \Gamma (z_i+w_i)}{ \Gamma (2w_i)} \frac{ \Gamma (\eta +w_i) \Gamma (\mu +w_i) \Gamma (w_i-\mu )}{ \Gamma (\eta +z_i) \Gamma (z_i+\mu ) \Gamma (z_i-\mu )}e^{t(F(z_i) -F(w_i))- t^{1/3} by(z_i-w_i)}.\nonumber \\ \end{aligned}$$

(C.1)

where $F(z)=G(z)+za_{\theta }$ and the function G is defined as in (1.19). Choosing

$$\begin{aligned} x_{\theta }=\frac{2 \psi _1(\theta )-\psi _1(\theta +\mu )-\psi _1(\theta -\mu )}{\psi _1(\theta +\mu )-\psi _1(\theta -\mu )}, \qquad a_{\theta }=-G'(\theta ), \end{aligned}$$

as in Conjecture 1.16 implies that $\theta$ is is a double critical point of the function F, i.e. $F'(\theta )=F''(\theta )=0$, so that by Taylor expansion,

$$\begin{aligned} F(z) = b_{\theta }^3 \frac{(z-\theta )^3}{3}+O\left( (z-\theta )^4\right) ,\end{aligned}$$

(C.2)

where $b_{\theta }$ is defined in Conjecture 1.16. Fix some $\theta >\mu$, and let us assume that we may deform the contours for the variables $w_i$ (resp. for the variables $z_i$) to some contour ${\mathcal {C}}$ (resp. some contour ${\mathcal {D}}$) going through $\theta$, in such a way that $z\mapsto \mathfrak {Re}[G(z)]$ attains its minimum (resp. maximum) along ${\mathcal {C}}$ (resp. ${\mathcal {D}}$) precisely at $z=\theta$. We further assume that $z\mapsto \mathfrak {Re}[G(z)]$ grows (resp. decays) sufficiently fast as $\vert z-\theta \vert$ increases. Then, we may approximate the integrals in (C.1) by their contribution in a neighborhood of $\theta$. This is the principle of Laplace’s method/steep descent analysis. Given the Taylor expansion (C.2), it is natural to rescale variables as

$$\begin{aligned} z_i=\theta +b_{\theta }^{-1} t^{-1/3}{\tilde{z}}_i, \qquad w_i=\theta +b_{\theta }^{-1} t^{-1/3} {\tilde{w}}_i, \end{aligned}$$

and we will call $\widetilde{{\mathcal {C}}}, \widetilde{{\mathcal {D}}}$ the rescaled contours. Under this change of variables, we have

$$\begin{aligned} tF(z_i) -t^{1/3} b_{\theta } yz_i =\frac{{\tilde{z}}_i^3}{3} -{\tilde{z}}_i y + O(t^{-1/3}). \end{aligned}$$

The other factors in (C.1) can be approximated as well, and taking into account some simplifications, one arrives at

$$\begin{aligned}{} & {} \lim _{t\rightarrow \infty } {\mathbb {E}}\left[ F_{\mu +\eta } \left( -\zeta _t(y) {\textsf{P}}_{0,2t}(x_{\theta }t,1)/4 \right) \right] \nonumber \\{} & {} \quad =1+\sum _{\ell =1}^{\infty } \int _{\widetilde{{\mathcal {C}}}}\frac{d{\tilde{w}}_1}{2{\textbf{i}}\pi } \dots \int _{\widetilde{{\mathcal {C}}}}\frac{d{\tilde{w}}_{\ell }}{2{\textbf{i}}\pi } \int _{\widetilde{{\mathcal {D}}}}\frac{d{\tilde{z}}_1}{2{\textbf{i}}\pi } \dots \int _{\widetilde{{\mathcal {D}}}}\frac{d{\tilde{z}}_{\ell }}{2{\textbf{i}}\pi } \det \left[ \frac{1}{{\tilde{z}}_i-{\tilde{w}}_j} \right] \prod _{i=1}^{\ell } \frac{1}{{\tilde{w}}_i-{\tilde{z}}_i} \frac{e^{\frac{{\tilde{z}}_i^3}{3}-y {\tilde{z}}_i}}{e^{\frac{{\tilde{w}}_i^3}{3}-y {\tilde{w}}_i}}.\nonumber \\ \end{aligned}$$

(C.3)

It is plausible that the contour $\widetilde{{\mathcal {C}}}$ (resp. $\widetilde{{\mathcal {D}}}$) can be deformed to become the union of two semi infinite rays going to infinity in directions $e^{\pm 2{\textbf{i}}\pi /3}$ (resp. $e^{\pm {\textbf{i}}\pi /3}$), in such a way that $\widetilde{{\mathcal {C}}}$ and $\widetilde{{\mathcal {D}}}$ do not intersect. Then, the right hand side of (C.3) would be exactly the Fredholm determinant expansion (see (A.2)) of the Tracy–Widom GUE distribution

$$\begin{aligned} F_\textrm{GUE}(y):= \det (I-K_{\textrm{Ai}})_{L^2(y,+\infty )}, \end{aligned}$$

(C.4)

where the Airy kernel is defined by

$$\begin{aligned} K_{\textrm{Ai}}(x,y) = \int _{\widetilde{{\mathcal {C}}}}\frac{d w}{2{\textbf{i}}\pi }\int _{\widetilde{{\mathcal {D}}}}\frac{d z}{2{\textbf{i}}\pi } \frac{1}{ z- w} \frac{e^{\frac{ z^3}{3}-x z}}{e^{\frac{w^3}{3}-y w}}. \end{aligned}$$

Using Lemma 4.6, this explains why we expect that

$$\begin{aligned} \lim _{t \rightarrow \infty } {\mathbb {P}}\left( \frac{\log \mathsf P_{0,2t}(x_{\theta } t, 1)-a_{\theta } t}{b_{\theta } t^{1/3}}\leqslant y \right) =F_\textrm{GUE}( y). \end{aligned}$$