Weakly Asymmetric Exclusion and KPZ

Quastel, Jeremy

doi:10.1142/9789814324359_0147

Cited by 5 publications

(7 citation statements)

References 30 publications

(42 reference statements)

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“…Then, choosing K such that ∞ k=K e −δk η < ρ/2 yields (32). A similar argument leads to (33). This completes the proof of the lemma.…”

Section: Equilibrium Casesupporting

confidence: 71%

“…In the line of specific first passage percolation models and interacting particle systems, a few explicitly solvable models were discovered, and they allow for detailed descriptions of new scaling limits and statistics. We namely mention Brownian queues [30], log-gamma polymer [35], KPZ equation [23,33]. However, the theory of universality classes does not explain the localization phenomena.…”

Section: Directed Polymers and Localizationmentioning

confidence: 99%

“…We will use the same method as in the section 4.1 to prove (46). With Lemma 6 at hand, we only need here to control the tail of sums as in (32) and (33). Define the minimum location of W .…”

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confidence: 99%

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Localization in log-gamma polymers with boundaries

Comets

Nguyen

2015

Probab. Theory Relat. Fields

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Consider the directed polymer in one space dimension in log-gamma environment with boundary conditions, introduced by Seppäläinen [35]. In the equilibrium case, we prove that the end point of the polymer converges in law as the length increases, to a density proportional to the exponent of a zero-mean random walk. This holds without space normalization, and the mass concentrates in a neighborhood of the minimum of this random walk. We have analogous results out of equilibrium as well as for the middle point of the polymer with both ends fixed. The existence and the identification of the limit relies on the analysis of a random walk seen from its infimum.

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“…Then, choosing K such that ∞ k=K e −δk η < ρ/2 yields (32). A similar argument leads to (33). This completes the proof of the lemma.…”

Section: Equilibrium Casesupporting

confidence: 71%

Section: Directed Polymers and Localizationmentioning

confidence: 99%

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Localization in log-gamma polymers with boundaries

Comets

Nguyen

2015

Probab. Theory Relat. Fields

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“…Experimental evidence of the relevance of that model has been obtained in a liquid crystal undergoing a phase transition [55]. This subject has generated many more works than could be summed up here, and the reader can find more information in reviews such as [56][57][58][59].…”

Section: Brief Overview Of the Asep's Family Treementioning

confidence: 98%

The physicist's companion to current fluctuations: one-dimensional bulk-driven lattice gases

Lazarescu¹

2015

J. Phys. A: Math. Theor.

102

156

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One of the main features of statistical systems out of equilibrium is the currents they exhibit in their stationary state: microscopic currents of probability between configurations, which translate into macroscopic currents of mass, charge, etc. Understanding the general behaviour of these currents is an important step towards building a universal framework for non-equilibrium steady states akin to the Gibbs-Boltzmann distribution for equilibrium systems. In this review, we consider onedimensional bulk-driven particle gases, and in particular the asymmetric simple exclusion process (ASEP) with open boundaries, which is one of the most popular models of one-dimensional transport. We focus, in particular, on the current of particles flowing through the system in its steady state, and on its fluctuations. We show how one can obtain the complete statistics of that current, through its large deviation function, by combining results from various methods: exact calculation of the cumulants of the current, using the integrability of the model ; direct diagonalisation of a biased process in the limits of very high or low current ; hydrodynamic description of the model in the continuous limit using the macroscopic fluctuation theory (MFT). We give a pedagogical account of these techniques, starting with a quick introduction to the necessary mathematical tools, as well as a short overview of the existing works relating to the ASEP. We conclude by drawing the complete dynamical phase diagram of the current. We also remark on a few possible generalisations of these results.

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“…Some of these conjectures have been verified for a handful of exactly solvable models. (See surveys [16,52,68,70].) In dimensions 3+1 and higher, high temperature behavior of directed polymers has been proved to be diffusive [15], but otherwise conjectures beyond 1+1 dimension are murky.…”

Section: Introductionmentioning

confidence: 99%

Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

Georgiou

Rassoul-Agha

Seppäläinen

2016

Commun. Math. Phys.

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We discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium: namely, the limiting time constant for last-passage percolation and the limiting free energy for directed polymers. The results are valid for models in arbitrary dimension, steps of the admissible paths can be general, the environment process is ergodic under spatial translations, and the potential accumulated along a path can depend on the environment and the next step of the path. The variational formulas come in two types: one minimizes over gradient-like cocycles, and another one maximizes over invariant measures on the space of environments and paths. Minimizing cocycles can be obtained from Busemann functions when these can be proved to exist. The results are illustrated through 1+1 dimensional exactly solvable examples, periodic examples, and polymers in weak disorder.F. Rassoul-Agha and N.

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Weakly Asymmetric Exclusion and KPZ

Cited by 5 publications

References 30 publications

Localization in log-gamma polymers with boundaries

Localization in log-gamma polymers with boundaries

The physicist's companion to current fluctuations: one-dimensional bulk-driven lattice gases

Variational Formulas and Cocycle solutions for Directed Polymer and Percolation Models

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