Oriented first passage percolation in the mean field limit

Kistler, Nicola; Schertzer, Adrien; Schmidt, Marius A.

doi:10.1214/18-bjps422

Cited by 4 publications

(9 citation statements)

References 6 publications

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“…The "law of large numbers" (4.1.3) naturally raises questions on fluctuations and weak limits, and calls for a description of the paths with minimal weight. As a first step towards this goal we presented in [37] an alternative, "modern" approach to (4.1.3) much inspired by the recent advances in the study of Derrida's random energy models (see [35] and references therein) and which relies on the hierarchical approximation to the FPP. In this companion paper we bring the approach to completion by establishing the full limiting picture, i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The former is rather straightforward: it only requires tail-estimates which we now state for they will be constantly used throughout the paper. (The simple proof may be found in [37,Lemma 5]). Lemma 19.…”

Section: Strategy Of Proofmentioning

confidence: 99%

“…In a first part of this Ph.D. thesis, which has lead to the publication [37], we propose a streamlined proof of the Fill-Pemantle theorem by means of the multiscale refinement of the second moment method [35], which, most importantly, establishes a neat point of contact between the FPP on the hypercube and hierarchical structures. This fresh take on the subject has been absolutely instrumental in our second contribution to the subject [38] which settles the full limiting picture / extremal process.…”

Section: Oriented Fpp On the Hypercubementioning

confidence: 99%

“…Die führende Ordnung des Grundzustands wurde von Aldous vermutet und vor mehr als drei Jahrzehnten von Fill und Pemantle bewiesen: Im ersten Teil dieser Arbeit schlagen wir einen einfacheren und intuitiveren Beweis des Fill-Pemantle-Theorems durch die multiscale refinement of the second moment method aus [35] vor, die in [37] erschienen ist. Vor allem hat diese Arbeit eine Verbindung zwischen der FPP auf dem Hyperwürfel und hierarchischen Strukturen hergestellt.…”

Section: Gerichtete Fpp Auf Dem Hyperwürfelunclassified

“…This chapter is up to minor changes published in [37]. The Poisson clumping heuristic has lead Aldous to conjecture the value of the oriented first passage percolation on the hypercube in the limit of large dimensions.…”

Section: Chapter 3 Oriented Fpp I: the First Ordermentioning

confidence: 99%

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First passage percolation in the mean field limit

Kistler¹,

Schertzer²,

Schmidt³

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Strategy Of Proofmentioning

confidence: 99%

Section: Oriented Fpp On the Hypercubementioning

confidence: 99%

Section: Gerichtete Fpp Auf Dem Hyperwürfelunclassified

Section: Chapter 3 Oriented Fpp I: the First Ordermentioning

confidence: 99%

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First passage percolation in the mean field limit

Kistler¹,

Schertzer²,

Schmidt³

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Oriented first passage percolation in the mean field limit, 2. The extremal process

Kistler¹,

Schertzer²,

Schmidt³

2020

Ann. Appl. Probab.

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Undirected Polymers in Random Environment: path properties in the mean field limit.

Kistler¹,

Schertzer²

2022

ALEA

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We consider the problem of undirected polymers (tied at the endpoints) in random environment, also known as the unoriented first passage percolation on the hypercube, in the limit of large dimensions. By means of the multiscale refinement of the second moment method we obtain a fairly precise geometrical description of optimal paths, i.e. of polymers with minimal energy. We give the distribution of the backsteps as a function of the Hamming distance from the origin. The picture which emerges can be loosely summarized as follows. The energy of the polymer is, to first approximation, uniformly spread along the strand. The polymer's bonds carry however a lower energy than in the directed setting, and are reached through the following geometrical evolution. Close to the origin, the polymer proceeds in oriented fashion -it is thus as stretched as possible. The tension of the strand decreases however gradually, with the polymer allowing for more and more backsteps as it enters the core of the hypercube. Backsteps, although increasing the length of the strand, allow the polymer to connect reservoirs of energetically favorable edges which are otherwise unattainable in a fully directed regime. These reservoirs lie at mesoscopic distance apart, but in virtue of the high dimensional nature of the ambient space, the polymer manages to connect them through approximate geodesics with respect to the Hamming metric: this is the key strategy which leads to an optimal energy/entropy balance. Around halfway, the mirror picture sets in: the polymer tension gradually builds up again, until full orientedness close to the endpoint. The approach yields, as a corollary, a constructive proof of the result by Martinsson [Ann. Appl. Prob. 26 (2016), Ann. Prob. 46 (2018] concerning the leading order of the ground state (or first passage percolation).In memory of Dima Ioffe.

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Oriented first passage percolation in the mean field limit

Cited by 4 publications

References 6 publications

First passage percolation in the mean field limit

First passage percolation in the mean field limit

Oriented first passage percolation in the mean field limit, 2. The extremal process

Undirected Polymers in Random Environment: path properties in the mean field limit.

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