Hausdorff dimensions for shared endpoints of disjoint geodesics in the directed landscape

Abstract

Within the Kardar-Parisi-Zhang universality class, the space-time Airy sheet is conjectured to be the canonical scaling limit for last passage percolation models. In recent work [27] of Dauvergne, Ortmann, and Virág, this object was constructed and, upon a parabolic correction, shown to be the limit of one such model: Brownian last passage percolation. The limit object without parabolic correction, called the directed landscape, admits geodesic paths between any two space-time points (x, s) and (y, t) with s < t. In this article, we examine fractal properties of the set of these paths. Our main results concern exceptional endpoints admitting disjoint geodesics. First, we fix two distinct starting locations x1 and x2, and consider geodesics traveling (x1, 0) → (y, 1) and (x2, 0) → (y, 1). We prove that the set of y ∈ R for which these geodesics coalesce only at time 1 has Hausdorff dimension one-half. Second, we consider endpoints (x, 0) and (y, 1) between which there exist two geodesics intersecting only at times 0 and 1. We prove that the set of such (x, y) ∈ R 2 also has Hausdorff dimension one-half.The proofs require several inputs of independent interest, including (i) connections to the so-called difference weight profile studied in [10]; and (ii) a tail estimate on the number of disjoint geodesics starting and ending in small intervals. The latter result extends the analogous estimate proved for the prelimiting model in [40].

The stationary horizon and semi-infinite geodesics in the directed landscape

“…The set of exceptional pairs of points between which there is a non-unique geodesic in the directed landscape L was studied in [BGH22]. Their approach relied on [BGH21] which studied the random nondecreasing function z → L(y, s; z, t) − L(x, s; z, t) for fixed x < y and s < t. This process is locally constant except on an exceptional set of Hausdorff dimension 1 2 .…”

“…Their approach relied on [BGH21] which studied the random nondecreasing function z → L(y, s; z, t) − L(x, s; z, t) for fixed x < y and s < t. This process is locally constant except on an exceptional set of Hausdorff dimension 1 2 . From here [BGH22] showed that for fixed s < t and x < y, the set of z ∈ R such that there exist disjoint geodesics from (x, s) to (z, t) and from (y, s) to (z, t) is exactly the set of local variation of the function z → L(x, s; z, t) − L(y, s; z, t), and therefore has Hausdorff dimension 1 2 . Going further, they showed that for fixed s < t, the set of pairs (x, y) ∈ R 2 such that there exist two disjoint geodesics from (x, s) to (y, t) also has Hausdorff dimension 1 2 , almost surely.…”

Infinite geodesics, competition interfaces and the second class particle in the scaling limit

“…Geodesics of L have properties that hold almost surely in the underlying randomness, some of which are listed below as they will be used throughout our discussion. These are derived from [3,9,10].…”

“…For every compact subset K ⊂ R 2 , there is an > 0 such that if g and g are two geodesics whose graphs lie in K and are within distance , then there is a common time t such that g(t) = g (t). See [3,Theorem 1.18].…”

Non-uniqueness times for the maximizer of the KPZ fixed point

“…This paper and [CHHM21] are also not the first to study fractal geometry in a KPZ limit. For example, [BGH21,BGH22] use Brownian absolute continuity results to study fractal geometry in the directed landscape. One major difference between the problem studied here and in [CHHM21] and those previous works is that here, we are trying to understand what happens as the time coordinate changes.…”