Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Lower bounds for fluctuations in first-passage percolation for general distributions

Michael Damron, Jack Hanson, Christian Houdré, and Chen Xu

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Abstract

In first-passage percolation (FPP), one assigns i.i.d. weights to the edges of the cubic lattice $\mathbb{Z}^{d}$ and analyzes the induced weighted graph metric. If $T(x,y)$ is the distance between vertices $x$ and $y$, then a primary question in the model is: what is the order of the fluctuations of $T(0,x)$? It is expected that the variance of $T(0,x)$ grows like the norm of $x$ to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order $\log \|x\|$. This result was found in the ’90s and there has not been any improvement since. In this paper, we address the problem of getting stronger fluctuation bounds: to show that $T(0,x)$ is with high probability not contained in an interval of size $o(\log \|x\|)^{1/2}$, and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of “hi-mode” (large).

Résumé

En percolation de premier passage (PPP), on attribue des poids i.i.d. aux arêtes du réseau cubique $\mathbb{Z}^{d}$ et analyse la métrique de graphe induite. Si $T(x,y)$ dénote la distance entre les sommets $x$ et $y$, une question fondamentale est de trouver l’ordre des fluctuations de $T(0,x)$. Il est escompté que la variance de $T(0,x)$ croît comme la norme, $\|x\|$, de $x$ à une puissance strictement inférieure à $1$, mais les meilleures bornes inférieures disponibles à ce jour (et seulement pour $d=2$) sont d’ordres logarithmiques. Ce résultat a été démontré dans les années 90 et il a connu peu d’amélioration depuis. Dans ce papier, nous abordons le problème d’obtenir des bornes inférieures plus strictes, en montrant qu’avec très grande probabilité, la distance $T(0,y)$ n’est pas contenue dans un interval de taille $o(\log \|x\|)^{1/2}$. Un résultat similaire est aussi valide en percolation de dernier passage (PDP) dans des cylindres minces. Ce type de résultats qui n’avait été obtenu que pour des classes particulières de poids est ici démontré en toute généralité. Les (nouvelles) méthodes développées ici consistent à induire une fluctuation du nombre d’arêtes dans une boîte dont les poids sont en “mode-haut.”

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1336-1357.

Dates
Received: 2 February 2019
Revised: 7 May 2019
Accepted: 21 May 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1584345640

Digital Object Identifier
doi:10.1214/19-AIHP1004

Mathematical Reviews number (MathSciNet)
MR4076786

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
First-passage percolation Fluctuations Small-ball probability

Citation

Damron, Michael; Hanson, Jack; Houdré, Christian; Xu, Chen. Lower bounds for fluctuations in first-passage percolation for general distributions. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1336--1357. doi:10.1214/19-AIHP1004. https://projecteuclid.org/euclid.aihp/1584345640


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