Existence of generalized Busemann functions and Gibbs measures for random walks in random potentials

TL;DR

This work develops a unified framework for generalized Busemann functions and Gibbs (DLR) measures for random walks in random potentials (RWRP) on $\mathbb{Z}^d$, capturing directed polymers, percolation models, and elliptic RWRE in both static and dynamic environments. The authors construct generalized Busemann functions $B^{\mathcal{A},\beta,m}$ via a hybrid approximation that blends point-to-hyperplane and point-to-level perspectives, under mild ergodicity and moment hypotheses, and show how these yield infinite-volume Gibbs measures $\textup{Q}_{x}^{\mathcal{A},\beta,m,\omega}$ consistently linked to restricted/unrestricted point-to-point measures and to the limiting free energy $\Lambda^{\beta}$. They establish that these cocycles satisfy recovery, cocycle, and covariance properties, and that the induced semi-infinite paths are directed into specific faces of the limit shape with well-defined large deviation principles for their velocity; the mean vectors $\mathbf{m}(B)$ align with the Legendre–Fenchel dual of the shape function. This Doob $h$-transform viewpoint connects generalized Busemann functions to harmonic functions, Martin boundary theory, and stochastic homogenization, offering a robust framework for analyzing semi-infinite geodesics and Gibbs measures in a broad class of random media. The results provide a versatile toolkit for understanding long-term path structure, DP/RWRE duality, and asymptotic velocity phenomena in complex random environments.

Abstract

We establish the existence of generalized Busemann functions and Gibbs-Dobrushin-Landford-Ruelle measures for a general class of lattice random walks in random potential with finitely many admissible steps. This class encompasses directed polymers in random environments, first- and last-passage percolation, and elliptic random walks in both static and dynamic random environments in all dimensions and with minimal assumptions on the random potential.

Figures (4)

• Figure 2.1: The thicker blue curves show plots of the approximation of $-\Lambda^{1}_{\mathop{\mathrm{res}}\nolimits}(t)$ from Example \ref{['ex:nRWRE']} by $-N^{-1}F_{\mathbf{0}, \lfloor Nt\rfloor, N}^{1}$ for $t\in[-1,1]$ in simulations. In the recurrent example in Figure \ref{['fig:RWRErec']}, $-\Lambda^{1}_{\mathop{\mathrm{res}}\nolimits}$ is differentiable and strictly convex. In the transient example with $0$ velocity in Figure \ref{['fig:RWREtv0']}, $-\Lambda^{1}_{\mathop{\mathrm{res}}\nolimits}$ is strictly convex and differentiable except at $0$. In the ballistic marginal nestling example in Figure \ref{['fig:RWREmnest']}, $-\Lambda^{1}_{\mathop{\mathrm{res}}\nolimits}$ has a flat segment and a linear segment, meeting at a corner at $0$. Elsewhere, it is differentiable and strictly convex. (The ballistic nestling case looks qualitatively the same.) In the non-nestling example in Figure \ref{['fig:RWREnnest']}, $-\Lambda^{1}_{\mathop{\mathrm{res}}\nolimits}$ has two linear segments that meet at a corner at $0$. Elsewhere, it is differentiable and strictly convex. The thinner red piecewise linear graphs show plots of $-\Lambda^{1}(t)$ for $t\in[-1,1]$. In all cases, $\Lambda^{1}(t)=0$ for $t\ge0$. In Figure \ref{['fig:RWRErec']}, $\Lambda^{1}(t)=0$ for $t\le0$ as well. In Figures \ref{['fig:RWREtv0']} and \ref{['fig:RWREmnest']}, the slope of $-\Lambda^{1}$ on $t<0$ matches the slope of the linear segment of $-\Lambda^{1}_{\mathop{\mathrm{res}}\nolimits}$ left of $0$. In Figure \ref{['fig:RWREnnest']}, $-\Lambda^{1}$ is tangent to $-\Lambda^{1}_{\mathop{\mathrm{res}}\nolimits}$ at a unique point in $(-1,0)$.
• Figure 2.2: A simulation of the ball of radius 75, $\{\xi:-\Lambda^{\infty}(\xi)\le75\}$, in Example \ref{['ex:FPPforb']} obtained by taking sub-level sets of the average of 100 samples of passage times from the origin. Both figures use the same data, so the pane on the right is the figure on the left viewed from below (with different coloring). We expect that $\Lambda^{\infty}$ is continuous on $\mathcal{C}$.
• Figure 3.1: The different panels depict various possible cases of $\mathcal{R}$ and $\mathcal{U}$. In each panel, the large balls represent $z\in\mathcal{R}$ that are extreme in $\mathcal{U}$, the small ball is $\mathbf{0}$, and $\mathcal{U}$ is the lightly shaded figure. Left to right: $\exists\widehat{u}:\forall x\in\mathcal{U},\ x\cdot\widehat{u}=1$; $\mathbf{0}\notin\mathcal{U}$; $\mathbf{0}\in\mathop{\mathrm{ext}}\nolimits\mathcal{U}$; $\mathbf{0}\in\mathcal{U}\setminus\mathrm{ri\,}\mathcal{U}$; $\mathbf{0}\in\mathrm{ri\,}\mathcal{U}$.
• Figure 3.2: The different panels depict various possible cases of the super-level sets $B_t=\{\xi\in\mathcal{C}:\Lambda^{\beta,\mathop{\mathrm{usc}}\nolimits}(\xi)\ge t\}$, which are depicted darkly shaded. When $\mathbf{0}\in\mathrm{ri\,}\mathcal{U}$, the cone $\mathcal{C}$ is the whole of $\mathbb{R}^d$. When $\mathbf{0}\notin\mathrm{ri\,}\mathcal{U}$, the cone $\mathcal{C}$ is depicted lightly shaded. These cones are unbounded but truncated in the drawings and hence appear as pyramids. The faces of each of these cones are the cone itself, the two-dimensional faces that are truncated in the drawing and appear as triangles on the side of the pyramid, the lines that are the boundaries of the two-dimensional faces, and the point $\mathbf{0}$. The super-level sets on the first row are all bounded. Left to right: $\mathbf{0}\in\mathrm{ri\,}\mathcal{U}$, $t<0$, and $\Lambda^{\beta,\mathop{\mathrm{usc}}\nolimits}$ is differentiable; $\mathbf{0}\in\mathrm{ri\,}\mathcal{U}$, $t<0$, and $B_t$ is polygonal; $\mathbf{0}\notin\mathcal{U}$, $\Lambda^{\beta,\mathop{\mathrm{usc}}\nolimits}(\xi)<0$$\forall\xi\ne\mathbf{0}$, $t<0$, and $B_t$ is polygonal. The shapes on the second row are all unbounded and are truncated in the drawings. Left to right: $\mathbf{0}\notin\mathcal{U}$, $\Lambda^{\beta,\mathop{\mathrm{usc}}\nolimits}$ takes both $<0$ and $\ge0$ values, $t<0$, and $B_t$ is polygonal; $\mathbf{0}\notin\mathcal{U}$, $\Lambda^{\beta,\mathop{\mathrm{usc}}\nolimits}$ takes both $>0$ and $<0$ values, $t>0$, and $\Lambda^{\beta,\mathop{\mathrm{usc}}\nolimits}$ is differentiable; $\mathbf{0}\notin\mathcal{U}$, $\Lambda^{\beta,\mathop{\mathrm{usc}}\nolimits}(\xi)>0$$\forall\xi\ne\mathbf{0}$, $t>0$, and $\Lambda^{\beta,\mathop{\mathrm{usc}}\nolimits}$ is differentiable.

Theorems & Definitions (110)

• Remark 2.1
• Remark 2.2
• Example 2.3: Edge and vertex weights
• Example 2.4: Product environment
• Example 2.5
• Definition 2.6
• Remark 2.7
• Remark 2.8
• Example 2.9: Nearest-neighbor RWRE on $\mathbb{Z}$
• Remark 2.10