Coalescence of geodesics in exactly solvable models of last passage percolation

Basu, Riddhipratim; Hoffman, Christopher; Sly, Allan

doi:10.1063/1.5093799

Cited by 67 publications

(123 citation statements)

References 44 publications

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“…A similar conceptual approach governs many ideas in [6,7], where the slow bond conjecture for the totally asymmetric exclusion process, concerning the macroscopic effects of slightly attenuating passage of particles through the origin, is proved, and the associated invariant measures determined. Such a probabilistic technique is also encountered in [5]. Indeed, in [5,Theorem 2], an assertion similar to, and in fact slightly stronger than, Theorem 1.5 has been proved for exponential last passage percolation: in essence, the assertion in the third bullet point in the preceding discussion has been verified for α = 1.…”

Section: Connections: Probabilistic Tools Geodesics and Coalescencementioning

confidence: 91%

“…Such a probabilistic technique is also encountered in [5]. Indeed, in [5,Theorem 2], an assertion similar to, and in fact slightly stronger than, Theorem 1.5 has been proved for exponential last passage percolation: in essence, the assertion in the third bullet point in the preceding discussion has been verified for α = 1.…”

Section: Connections: Probabilistic Tools Geodesics and Coalescencementioning

confidence: 91%

“…Moreover, Pimentel conjectured that the probability that coalescence has failed to occur by time

r ℓ^{3 / 2}

has tail

r^{- 2 / 3}

. In the context of finite geodesics, has proved an upper bound on this failure probability of the form

r^{- c}

for some positive

c > 0

, 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 .…”

Section: Introductionmentioning

confidence: 99%

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

Hammond

2019

Proc. London Math. Soc.

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In last passage percolation models lying in the KPZ universality class, long maximizing paths have a typical deviation from the linear interpolation of their endpoints governed by the two‐thirds power of the interpolating distance. This two‐thirds power dictates a choice of scaled coordinates, in which these maximizers, now called polymers, cross unit distances with unit‐order fluctuations. In this article, we consider Brownian last passage percolation in these scaled coordinates, and prove that the probability of the presence of k disjoint polymers crossing a unit‐order region while beginning and ending within a short distance ε of each other is bounded above by ε(k2−1)/2+o(1). This result, which we conjecture to be sharp, yields understanding of the uniform nature of the coalescence structure of polymers, and plays a foundational role in Hammond (Forum Math. Pi 7 (2019) e2, 69) in proving comparison on unit‐order scales to Brownian motion for polymer weight profiles from general initial data. The present paper also contains an on‐scale articulation of the two‐thirds power law for polymer geometry: polymers fluctuate by ε2/3 on short scales ε.

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Section: Connections: Probabilistic Tools Geodesics and Coalescencementioning

confidence: 91%

Section: Connections: Probabilistic Tools Geodesics and Coalescencementioning

confidence: 91%

“…Moreover, Pimentel conjectured that the probability that coalescence has failed to occur by time

r ℓ^{3 / 2}

has tail

r^{- 2 / 3}

. In the context of finite geodesics, has proved an upper bound on this failure probability of the form

r^{- c}

for some positive

c > 0

Section: Introductionmentioning

confidence: 99%

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

Hammond

2019

Proc. London Math. Soc.

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“…These are exactly solvable models, for which certain exact distributional formulas are available, and the derivations of these formulas typically employ deep machinery from algebraic combinatorics or random matrix theory. It is interesting to study geometric properties of universal KPZ objects by approaches that, while they are reliant on certain integrable inputs, are probabilistic in flavour: for example, [4], [2] and [3] are recent results and applications concerning geometric properties of last passage percolation paths.…”

Section: Introductionmentioning

confidence: 99%

Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation

Hammond¹,

Sarkar²

2020

Electron. J. Probab.

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In last passage percolation models, the energy of a path is maximized over all directed paths with given endpoints in a random environment, and the maximizing paths are called geodesics. The geodesics and their energy can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order. Here we consider Poissonian last passage percolation, a model lying in the KPZ universality class, and refer to scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most t 2/3 log t −1 1/3 . The power of one-third in the logarithm may be expected to be sharp and in a related problem we show that it is: among polymers in the unit box whose endpoints have vertical separation t (and a horizontal separation of the same order), the maximum transversal fluctuation has order t 2/3 log t −1 1/3 . Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at (0, 0) and the other is varied vertically over (0, z), z ∈ [1, 2], the resulting random weight profile has sharp modulus of continuity of order t 1/3 log t −1 2/3 . In this way, we identify exponent pairs of (2/3, 1/3) and (1/3, 2/3) in power law and polylogarithmic correction, respectively for polymer fluctuation, and polymer weight under vertical endpoint perturbation. The two exponent pairs describe [10,11,9] the fluctuation of the boundary separating two phases in subcritical planar random cluster models.2010 Mathematics Subject Classification. 82B23, 82C23 and 60K35.

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“…Chuck's work on first passage percolation has been and continues to be very influential. His results on the subject continue to be cited, and the ideas and methods developed by Chuck and coauthors have been used and extended in numerous papers (for a small selection, see [93,126,112,64,74,19,73,242]).…”

Section: Percolationmentioning

confidence: 99%

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

Camia

Stein

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Coalescence of geodesics in exactly solvable models of last passage percolation

Cited by 67 publications

References 44 publications

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

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

Modulus of continuity for polymer fluctuations and weight profiles in Poissonian last passage percolation

Probability Theory in Statistical Physics, Percolation, and Other Random Topics: The Work of C. Newman

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