Nonexistence of Bigeodesics in Planar Exponential Last Passage Percolation

Basu, Riddhipratim; Hoffman, Christopher; Sly, Allan

doi:10.1007/s00220-021-04246-0

Cited by 15 publications

(12 citation statements)

References 47 publications

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“…environments. The following result can be found as formula ( 3) and ( 4) in [15], and Theorem 2 in [13] for the lower bound in (3.2). Lemma 3.1.…”

Section: 1mentioning

confidence: 86%

“…The detailed analysis of last passage percolation using inputs from integrable probability together with probabilistic concepts falls into a more general class of recent articles. Precise estimates were achieved for example on the coalescence of geodesics and non-existence of bi-infinite geodesics [17,63,67], the correlation of geodesics and last passage times [8,10,11,12,14,15], and on the current and invariant measures for TASEP with a slow bond [16,18]. We will see in Sections 2 to 4 that several of these results have natural analogues in our setup for the TASEP on the circle.…”

Section: Introductionmentioning

confidence: 94%

“…These bounds are then transferred to periodic environments. In particular, we focus on moderate deviation estimates for last passage times and transversal fluctuations of flat geodesics; see also [15] for a similar approach in i.i.d. environments.…”

Section: Introductionmentioning

confidence: 99%

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Mixing times for the TASEP on the circle

Schmid¹,

Sly²

2022

Preprint

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We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a circle of length N with k particles. We show that the mixing time is of order N 2 min(k, N −k) −1/2 , and that the cutoff phenomenon does not occur. This confirms behavior which was separately predicted by Jara, Lacoin and Peres, and it is more broadly believed to hold for integrable models in the KPZ-universalty class. Our arguments rely on a connection to periodic last passage percolation with a detailed analysis of flat geodesics, as well as a novel random extension and time shift argument for last passage percolation.

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“…environments. The following result can be found as formula ( 3) and ( 4) in [15], and Theorem 2 in [13] for the lower bound in (3.2). Lemma 3.1.…”

Section: 1mentioning

confidence: 86%

Section: Introductionmentioning

confidence: 94%

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Mixing times for the TASEP on the circle

Schmid¹,

Sly²

2022

Preprint

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“…Even though one expects that the exponent 1/2 is universal for a large class of passage time distribution, our proof will fully exploit the specific integrability properties of exponential LPP. In contrast with the result of [9], which did not require exact solvability, and the recent results in [12,11,7,10] which only required the moderate deviation estimates for the last passage time (that are available for a number of other exactly solvable models…”

Section: Introductionmentioning

confidence: 86%

Connecting Eigenvalue Rigidity with Polymer Geometry: Diffusive Transversal Fluctuations under Large Deviation

Basu¹,

Ganguly²

2019

Preprint

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We consider the exactly solvable model of exponential directed last passage percolation on Z 2 in the large deviation regime. Conditional on the upper tail large deviation event U δ := {Tn ≥ (4 + δ)n} where Tn denotes the last passage time from (1, 1) to (n, n), we study the geometry of the polymer/geodesic Γn, i.e., the optimal path attaining Tn. We show that conditioning on U δ changes the transversal fluctuation exponent from the characteristic 2/3 of the KPZ universality class to 1/2, i.e., conditionally, the smallest strip around the diagonal that contains Γn has width n 1/2+o(1) with high probability. This sharpens a result of Deuschel and Zeitouni (1999) [19] who proved a o(n) bound on the transversal fluctuation in the context of Poissonian last passage percolation, and complements [9], where the transversal fluctuation was shown to be Θ(n) in the lower tail large deviation event. Our proof exploits the correspondence between last passage times in the exponential LPP model and the largest eigenvalue of the Laguerre Unitary Ensemble (LUE) together with the determinantal structure of the spectrum of the latter. A key ingredient in our proof is a sharp refinement of the large deviation result for the largest eigenvalue [26,37], using rigidity properties of the spectrum, which could be of independent interest.

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“…3. Two preprints (by Basu-Hoffman-Sly [7] and Balázs-Busani-Seppäläinen [6]) have recently been posted which state results of nonexistence of bigeodesics in a related model, exactly solvable last-passage percolation in two dimensions. In this model, weights are placed on the sites and must be exponentially distributed, one replaces the infimum in the definition of T with supremum, and one considers only oriented paths.…”

Section: Infinite Geodesics In Fppmentioning

confidence: 99%

Absence of backward infinite paths for first-passage percolation in arbitrary dimension

Brito¹,

Damron²,

Hanson³

2020

Preprint

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In first-passage percolation (FPP), one places nonnegative random variables (weights) (t e ) on the edges of a graph and studies the induced weighted graph metric. We consider FPP on Z d for d ≥ 2 and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifically, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture originating from a question of Furstenberg and most strongly supported for d = 2 that for continuously distributed i.i.d. weights, there a.s. are no bigeodesics. We provide the first progress on this question in general dimensions under no unproven assumptions. Our main result is that geodesic graphs, introduced in a previous paper of two of the authors, constructed in any deterministic direction a.s. do not contain doubly-infinite paths. As a consequence, one can construct random graphs of subsequential limits of point-to-hyperplane geodesics which contain no bigeodesics. This gives evidence that bigeodesics, if they exist, cannot be constructed in a translation-invariant manner as limits of point-to-hyperplane geodesics.

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Nonexistence of Bigeodesics in Planar Exponential Last Passage Percolation

Cited by 15 publications

References 47 publications

Mixing times for the TASEP on the circle

Mixing times for the TASEP on the circle

Connecting Eigenvalue Rigidity with Polymer Geometry: Diffusive Transversal Fluctuations under Large Deviation

Absence of backward infinite paths for first-passage percolation in arbitrary dimension

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