Let $T$ be a random ergodic pseudometric over $\mathbb R^d$. This setting generalizes the classical \emph{first passage percolation} (FPP) over $\mathbb Z^d$. We provide simple conditions on $T$, the decay of instant one-arms and exponential quasi-independence, that ensure the positivity of its time constants, that is almost surely, the pseudo-distance given by $T$ from the origin is asymptotically a norm. Combining this general result with previously known ones, we prove that
The known phase transition for Gaussian percolation in the case of fields with positive correlations with exponentially fast decayholds for Gaussian FPP, including the natural Bargmann-Fock model;
The known phase transition for Voronoi percolation also extends to the associated FPP;
The same happens for Boolean percolation for radii with exponential tails, a result which was known without this condition.
We prove the positivity of the constant for random continuous Riemannian metrics, including cases with infinite correlations in dimension $d=2$.
Finally, we show that the critical exponent for the one-arm, if exists, is bounded above by $d-1$. This holds forbond Bernoulli percolation, planar Gaussian fields, planar Voronoi percolation, and Boolean percolation with exponential small tails.
Recently, many results have been established drawing a parallel between Bernoulli percolation and models given by levels of smooth Gaussian fields with unbounded, strongly decaying correlation. In a previous work with D. Gayet , we started to extend these analogies by adapting the first basic results of classical first passage percolation in this new framework: positivity of the time constant and the ball-shape theorem. In the present paper, we present a proof inspired by Kesten of other basic properties of the new FPP model: an upper bound on the variance in the FPP pseudometric given by the …
There are various models of first passage percolation (FPP) in $\mathbb R^d$. We want to start a very general study of this topic. To this end we generalize the first passage percolation model on the lattice $\mathbb Z^d$ to $\mathbb R^d$ and adapt the results of \cite{boivin1990first} to prove a shape theorem for ergodic random pseudometrics on $\mathbb R^d$. A natural application of this result will be the study of FPP on random tessellations where a fluid starts in the zero cell and takes a random time to pass through the boundary of a cell into a neighbouring cell. We find that a tame rand…
We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on $\mathbb R^d$. We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geo…
This paper studies the first passage percolation (FPP) model: each edge in the cubic lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region $B(t)$, which consists of those vertices that can be reached from the origin within a time $t > 0$. Cox and Durrett showed the shape theorem for the percolation region, saying that the normalized region $B(t)/t$ converges to some limit shape $\mathcal{B}$. This paper introduces a general FPP model defined on crystal lattices, and shows the monotonicity of the limit shapes under covering maps, thereby…
We study first passage percolation on the configuration model (CM) having power-law degrees with exponent $τ\in [1,2)$. To this end we equip the edges with exponential weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal weight path, which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via the construction of an infinite limiting object describing the FPP problem in the densely connected core of the network. We consider two separate cases, n…
The main contribution of this paper is the development of a novel approach to multi-scale analysis that we believe can be used to analyse processes with non-equilibrium dynamics. Our approach will be referred to as \emph{multi-scale analysis with non-equilibrium feedback} and will be used to analyse a natural random growth process with competition on $\mathbb{Z}^d$ called \emph{first passage percolation in a hostile environment} that consists of two first passage percolation processes $FPP_1$ and $FPP_λ$ that compete for the occupancy of sites. Initially, $FPP_1$ occupies the origin and spread…
We consider first passage percolation (FPP) where the vertex weight is given by the exponential of two-dimensional log-correlated Gaussian fields. Our work is motivated by understanding the discrete analog for the random metric associated with \emph{Liouville quantum gravity} (LQG), which roughly corresponds to the exponential of a two-dimensional Gaussian free field (GFF). The particular focus of the present paper is an aspect of universality for such FPP among the family of log-correlated Gaussian fields. More precisely, we construct a family of log-correlated Gaussian fields, and show that …
We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight $e^{γ\mathcal R^N_z}$ for $γ>0$. We show that for $γ>0$ sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most $O(N^{1 - γ^2/10})$.
In this paper we explore first passage percolation (FPP) on the Erdős-Rényi random graph $G_n(p_n)$, where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when $np_n\to λ>1,$ we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to $λ/(λ-1)\log{n}$. Furthermore, we prove that the…
Rotationally invariant first passage percolation: Breaking the $n/\log n$ variance barrier
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）；Courant Institute of Mathematical Sciences, New York and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学柯朗数学科学研究所，上海纽约大学纽大-华东师大数学科学研究所）；Department of Mathematics, Princeton University（普林斯顿大学数学系）
Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in $\R^d$. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.
We investigate first passage percolation on inhomogeneous random graphs. The random graph model G(n,kappa) we study is the model introduced by Bollobás, Janson and Riordan, where each vertex has a type from a type space S and edge probabilities are independent, but depending on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal weight path, properly normalized follows a ce…
In first-passage percolation (FPP), one places nonnegative random variables (weights) $(t_e)$ on the edges of a graph and studies the induced weighted graph metric. We consider FPP on $\mathbb{Z}^d$ for $d \geq 2$ and analyze the geometric properties of geodesics, which are optimizing paths for the metric. Specifically, we address the question of existence of bigeodesics, which are doubly-infinite paths whose subpaths are geodesics. It is a famous conjecture originating from a question of Furstenberg and most strongly supported for $d=2$ that for continuously distributed i.i.d. weights, there …
We study the geometry of infinite random Boltzmann planar maps having weight of polynomial decay of order $k^{-2}$ for each vertex of degree $k$. These correspond to the dual of the discrete "stable maps" of Le Gall and Miermont [Scaling limits of random planar maps with large faces, Ann. Probab. 39, 1 (2011), 1-69] studied in [Budd & Curien, Geometry of infinite planar maps with high degrees, Electron. J. Probab. (to appear)] related to a symmetric Cauchy process, or alternatively to the maps obtained after taking the gasket of a critical $O(2)$-loop model on a random planar map. We show that…
First passage percolation (FPP) on $\mathbb{Z}^d$ or $\mathbb{R}^d$ is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of the geodesic, the shortest path between points. Since the latter, owing to its length minimization, traverses through atypically low values of the underlying noise variables, it is an important problem to quantify the disparity between the environment rooted at a point on the geodesic and the typical one. We investigate this in the context of $γ$-Liouville Quant…
