Bigeodesics in First-Passage Percolation

Abstract

In first-passage percolation, we place i.i.d. continuous weights at the edges of Z 2 and consider the weighted graph metric. A distance-minimizing path between points x and y is called a geodesic, and a bigeodesic is a doubly-infinite path whose segments are geodesics. It is a famous conjecture that almost surely, there are no bigeodesics. In the '90s, Licea-Newman showed that, under a curvature assumption on the "asymptotic shape," all infinite geodesics have an asymptotic direction, and there is a full measure set D ⊂ [0, 2π) such that for any θ ∈ D, there are no bigeodesics with one end directed in direction θ. In this paper, we show that there are no bigeodesics with one end directed in any deterministic direction, assuming the shape boundary is differentiable. This rules out existence of ground state pairs for the related disordered ferromagnet whose interface has a deterministic direction. Furthermore, it resolves the BenjaminiKalai-Schramm "midpoint problem" [5, p. 1976] under the extra assumption that the limit shape boundary is differentiable.

Thermodynamic Limit for Directed Polymers and Stationary Solutions of the Burgers Equation

“…Tightly connected to the thermodynamic limit results in [GRASY15] are results on the limits of ratios of partition functions. Logarithms of these limiting ratios are polymer counterparts of Busemann functions that compare actions of infinite geodesics to each other in zero temperature models such as first passage percolation (FPP), last passage percolation, or zero-viscosity Burgers equation, see [HN01], [CP12], [BCK14], [Bak16], [GRAS16], [GRS15], [DH14], [DH17] and [AHD15], which is a recent survey on FPP. In [GRAS16] and [RSY16] a variational approach to ratios of partition functions is described.…”

Exponents governing the rarity of disjoint polymers in Brownian last passage percolation

“…In the context of finite geodesics, has proved an upper bound on this failure probability of the form for some positive , and there seems to be promise in these techniques to obtain a sharper form of the result. Hoffman initiated very fruitful progress on infinite geodesics in first passage percolation by using Busemann functions, with significant geometric information emerging in . Busemann functions have also been studied in non‐integrable last passage percolation contexts: see .…”

Infinite geodesics, asymptotic directions, and Busemann functions in first-passage percolation

“…Recall that Licea and Newman showed that for "typical" directions θ -i.e., all θ in some set D Ď r0, 2πq -there is a.s. a unique θ-directed geodesic. We will describe work from [7] which extends these uniqueness statements using Busemann function techniques. In keeping with the above discussion, these results give not uniqueness of geodesics having some single direction θ, but rather uniqueness of geodesics directed within sectors (as produced by Theorem 7.2).…”