Variational Formula for the Time Constant of First‐Passage Percolation

Krishnan, A.

doi:10.1002/cpa.21648

Cited by 12 publications

(16 citation statements)

References 58 publications

(146 reference statements)

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“…This result can be viewed as a type of stochastic homogenization [37], where the effective Hamiltonian is given in (P). A similar stochastic homogenization result has been obtained recently for first passage percolation [27], though in that case the exact form of the effective Hamiltonian is unknown. The Hamilton-Jacobi equation (P) is also closely related to the conservation law for the hydrodynamic limit of TASEP [21], and in Section 1.2 we show a formal equivalence between the two continuum limits.…”

Section: Resultssupporting

confidence: 71%

Directed Last Passage Percolation with Discontinuous Weights

Calder

2014

J Stat Phys

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We prove that a directed last passage percolation model with discontinuous macroscopic (non-random) inhomogeneities has a continuum limit that corresponds to solving a Hamilton-Jacobi equation in the viscosity sense. This Hamilton-Jacobi equation is closely related to the conservation law for the hydrodynamic limit of the totally asymmetric simple exclusion process. We also prove convergence of a numerical scheme for the Hamilton-Jacobi equation and present an algorithm based on dynamic programming for finding the asymptotic shapes of maximal directed paths.

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Section: Resultssupporting

confidence: 71%

Directed Last Passage Percolation with Discontinuous Weights

Calder

2014

J Stat Phys

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“…We write Ω for the probability space and σ v : Ω → Ω for the shift that translates the random variables τ e by v. Define where (·, ·) is the standard inner product in R d . The following is the main result of [120]. (p, x).…”

Section: The Time Constant Through a Homogenization Problemmentioning

confidence: 87%

“…Another way to interpret the time constant was recently explored by Krishnan [120] in FPP and by Georgiou, Rassoul-Agha and Seppäläinen [85] in last-passage percolation. We briefly describe it here.…”

Section: The Time Constant Through a Homogenization Problemmentioning

confidence: 99%

50 Years of First-Passage Percolation

Auffinger

Damron

Hanson

2017

University Lecture Series

230

405

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We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2 + log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.

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“…A general theory that is a step towards universality can be found at the law of large numbers level [21,36,37,38] where a series of variational formulas for the limiting free energy density of polymer models and shape functions for last passage percolation where proven. A variational formula for the time constant in first passage percolation was proven in [29]. For two-dimensional last passage models with e 1 , e 2 admissible steps the analysis and results can be sharpened; early universal results on the shape near the edge were obtained in [32,7].…”

Section: Solvable Models Of Lattice Last Passage Percolation and Kpz mentioning

confidence: 99%

Order of the Variance in the Discrete Hammersley Process with Boundaries

Ciech

Georgiou

2019

J Stat Phys

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We discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope.We show that along characteristic directions the order of the variance of the last passage time is of order N 2/3 in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along any direction outside the cone, the order of the variance changes to O(N ) in the boundary model and to O(1) for the nonboundary model. This behaviour is the result of the two flat edges of the shape function.

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Variational Formula for the Time Constant of First‐Passage Percolation

Cited by 12 publications

References 58 publications

Directed Last Passage Percolation with Discontinuous Weights

Directed Last Passage Percolation with Discontinuous Weights

50 Years of First-Passage Percolation

Order of the Variance in the Discrete Hammersley Process with Boundaries

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