Let C_n be the origin-containing cluster in subcritical percolation on the lattice (1/n) Z^d, viewed as a random variable in the space Omega of compact, connected, origin-containing subsets of R^d, endowed with the Hausdorff metric delta. When d >= 2, and Gamma is any open subset of Omega, we prove: lim_{n \to \infty}(1/n) \log P(C_n \in Γ) = -\inf_{S \in Γ} λ(S) where lambda(S) is the one-dimensional Hausdorff measure of S defined using the {\em correlation norm}: ||u|| := \lim_{n \to \infty} - \frac{1}{n} \log P (u_n \in C_n) where u_n is u rounded to the nearest element of (1/n)Z^d. Given points a^1, >..., a^k in R^d, there are finitely many correlation-norm Steiner trees spanning these points and the origin. We show that if the C_n are each conditioned to contain the points a^1_n,..., a^k_n, then the probability that C_n fails to approximate one of these trees decays exponentially in n.
Consider first passage percolation with identical and independent weight distributions and first passage time ${\rm T}$. In this paper, we study the upper tail large deviations $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$, for $ξ>0$ and $x\neq 0$ with a time constant $μ$ and a dimension $d$, for weights that satisfy a tail assumption $ β_1\exp{(-αt^r)}\leq \mathbb P(τ_e>t)\leq β_2\exp{(-αt^r)}.$ When $r\leq 1$ (this includes the well-known Eden growth model), we show that the upper tail large deviation decays as $\exp{(-(2dξ+o(1))n)}$. When $1< r\leq d$, we find that the rate function can be naturally d…
We consider biased random walk on supercritical percolation clusters in $\Z^2$. We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive.
For first passage percolation on $\mathbb{Z}^2$ with i.i.d. bounded edge weights, we consider the upper tail large deviation event; i.e., the rare situation where the first passage time between two points at distance $n$, is macroscopically larger than typical. It was shown by Kesten (1986) that the probability of this event decays as $\exp (-Θ(n^2))$. However the question of existence of the rate function i.e., whether the log-probability normalized by $n^2$ tends to a limit, had remained open. We show that under some additional mild regularity assumption on the passage time distribution, the…
For the exactly solvable model of exponential last passage percolation on $\mathbb{Z}^2$, consider the geodesic $Γ_n$ joining $(0,0)$ and $(n,n)$ for large $n$. It is well known that the transversal fluctuation of $Γ_n$ around the line $x=y$ is $n^{2/3+o(1)}$ with high probability. We obtain the exponent governing the decay of the small ball probability for $Γ_{n}$ and establish that for small $δ$, the probability that $Γ_{n}$ is contained in a strip of width $δn^{2/3}$ around the diagonal is $\exp (-Θ(δ^{-3/2}))$ uniformly in high $n$. We also obtain optimal small deviation estimates for the …
We show quenched large deviations for the simple random walk on a certain class of percolations with long-range correlations. This class contains the supercritical Bernoulli percolations, the model considered by Drewitz, R'ath and Sapozhnikov and the random-cluster model up to the slab critical point. Our result is an extension of Kubota's result for the supercritical Bernoulli percolations. We also state a shape theorem for the chemical distance, which is an extension of Garet and Marchand's result for the supercritical Bernoulli percolations.
By adapting the renormalization techniques of Pisztora, we establish surface order large deviations estimates for FK-percolation on $\Z^2$ with parameter $q\geq 1$ and for the corresponding Potts models. Our results are valid up to the exponential decay threshold of dual connectivities which is widely believed to agree with the critical point.
This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.
Following the recent investigations of Baik and Suidan in \cite{baik2005gcl} and Bodineau and Martin in \cite{bodineau2005upl}, we prove large deviation properties for a last-passage percolation model in $\mathbb{Z}^{2}_{+}$ whose paths are close to the axis. The results are mainly obtained when the random weights are Gaussian or have a finite moment-generating function and rely, as in \cite{baik2005gcl} and \cite{bodineau2005upl}, on an embedding in Brownian paths and the KMT approximation. The study of the subexponential case completes the exposition.
Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. We prove that the lower large deviations are of surface order, and we prove the corresponding large deviation principle from below. This extends and improves previous large deviations results of Grimmett and Kesten (1984) obtained for boxes of particular orientation.
We consider planar stationary exponential Last Passage Percolation in the positive quadrant with boundary weights. For $ρ\in (0,1)$ and points $v_N=((1-ρ)^2 N,ρ^2 N)$ going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations for the passage time. For the case $ρ=\frac{1}{2}$ in the stationary model, we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time. Our arguments are based on …
In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${\rm T}$. Our main results prove that for any $ξ>0$ and $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$ decays as $\exp{(-(2dξ+o(1))n)}$ with a time constant $μ$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.
In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.
We study moderate deviations in the exponential corner growth model, both in the bulk setting and the increment-stationary setting. The main results are sharp right-tail bounds on the last-passage time and the exit point of the increment-stationary process. The arguments utilize calculations with the stationary version and a moment generating function identity due to E. Rains, for which we give a short probabilistic proof. As applications of the deviation bounds, we derive upper bounds on the speed of distributional convergence in the Busemann function and competition interface limits.
The chemical distance D(x,y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $μ$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{μ(x)}\notin (1-ε, 1+ε) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.
We present a large-deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the probability that the giant component occupies a fixed fraction of the graph while all other components are ``small.'' One consequence is an immediate derivation of the ``cavity'' formula for the fraction of vertices in the giant component. As a by-product of our analysis we compute the large-deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no …
We prove the sharpness of the phase transition for speed in the biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least 2, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias, such that, below this value, the speed is positive, and, above the value, it is zero. We identify the value of the critical bias explicitly, and, in the sub-ballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that ar…
We consider the Erdös--Rényi random graph $G_{n,p}$ and we analyze the simple irreversible epidemic process on the graph, known in the literature as bootstrap percolation. We give a quantitative version of some results by Janson et al. (2012), providing a fine asymptotic analysis of the final size $A_n^*$ of active nodes, under a suitable super-critical regime. More specifically, we establish large deviation principles for the sequence of random variables $\{\frac{n- A_n^*}{f(n)}\}_{n\geq 1}$ with explicit rate functions and allowing the scaling function $f$ to vary in the widest possible rang…
Let M_n denote the number of sites in the largest cluster in critical site percolation on the triangular lattice inside a box side length n. We give lower and upper bounds on the probability that M_n / E(M_n) > x of the form exp(- C x^(2/alpha)) for x > 1 and large n with alpha = 5/48 and C > 0. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions d > 2.
We consider the standard first passage percolation model in the rescaled lattice $\mathbb Z^d/n$ for $d\geq 2$ and a bounded domain $Ω$ in $\mathbb R^d$. We denote by $Γ^1$ and $Γ^2$ two disjoint subsets of $\partial Ω$ representing respectively the sources and the sinks, \textit{i.e.}, where the water can enter in $Ω$ and escape from $Ω$. A cutset is a set of edges that separates $Γ^1$ from $Γ^2$ in $Ω$, it has a capacity given by the sum of the capacities of its edges. Under some assumptions on $Ω$ and the distribution of the capacities of the edges, we already know a law of large numbers fo…
We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $Ω$ of boundary $Γ$ in $\mathbb{R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$, representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behaviour of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviat…
We study the speed of a biased random walk on a percolation cluster on $\Z^d$ in function of the percolation parameter $p$. We obtain a first order expansion of the speed at $p=1$ which proves that percolating slows down the random walk at least in the case where the drift is along a component of the lattice.
We consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of trees. A sufficient condition is established for Galton-Watson trees.
We consider the standard first passage percolation model in the rescaled lattice $\mathbb{Z}^d$ for $d\geq 2$ and a bounded domain $Ω$ in $\mathbb R ^d$. We denote by $Γ^1$ and $Γ^2$ two disjoint subsets of $\partial Ω$ representing respectively the source and the sink, i.e., where the water can enter in $Ω$ and escape from $Ω$. A maximal stream is a vector measure $\overrightarrowμ_n^{max}$ that describes how the maximal amount of fluid can enter through $Γ^1$ and spreads in $Ω$. Under some assumptions on $Ω$ and $G$, we already know a law of large number for $\overrightarrowμ_n^{max}$. The s…
We consider the standard first passage percolation in $\mathbb{Z}^{d}$ for $d\geq 2$ and we denote by $ϕ_{n^{d-1},h(n)}$ the maximal flow through the cylinder $]0,n]^{d-1} \times ]0,h(n)]$ from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension three: under some assumptions, $ϕ_{n^{d-1},h(n)} / n^{d-1}$ converges towards a constant $ν$. We look now at the probability that $ϕ_{n^{d-1},h(n)} / n^{d-1}$ is greater than $ν+ ε$ for some $ε>0$, and we show under some assumptions that this probability decays exponentially fast with the volume of the cylinder…
For biased random walk on the infinite cluster in supercritical i.i.d.\ percolation on $\Z^2$, where the bias of the walk is quantified by a parameter $β>1$, it has been conjectured (and partly proved) that there exists a critical value $β_c>1$ such that the walk has positive speed when $β<β_c$ and speed zero when $β>β_c$. In this paper, biased random walk on the infinite cluster of a certain translation invariant percolation process on $\Z^2$ is considered. The example is shown to exhibit the opposite behavior to what is expected for i.i.d.\ percolation, in the sense that it has a critical va…
We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster (SRWPC) on $\mathbb Z^d$ ($d\geq 2$). The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and {the vacant set of random interlacements} (for $d\geq 3$) and the level sets of the Gaussian free field ($d\geq 3$). Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffus…
For the last passage percolation (LPP) on $\mathbb{Z}^2$ with exponential passage times, let $T_{n}$ denote the passage time from $(1,1)$ to $(n,n)$. We investigate the law of iterated logarithm of the sequence $\{T_{n}\}_{n\geq 1}$; we show that $\liminf_{n\to \infty} \frac{T_{n}-4n}{n^{1/3}(\log \log n)^{1/3}}$ almost surely converges to a deterministic negative constant and obtain some estimates on the same. This settles a conjecture of Ledoux (J. Theor. Probab., 2018) where a related lower bound and similar results for the corresponding upper tail were proved. Our proof relies on a slight …
In this paper we will show how the results found in Cator and Pimentel 2009, about the Busemann functions in last-passage percolation, can be used to calculate the asymptotic distribution of the speed of a single second class particle starting from an arbitrary deterministic configuration which has a rarefaction fan, in either the totally asymetric exclusion process, or the Hammersley interacting particle process. The method will be to use the well known last-passage percolation description of the exclusion process and of the Hammersley process, and then the well known connection between secon…
We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster on $\Z^d$, $d\geq 2$.. We take the point of view of the moving particle and first prove a quenched LDP for the distribution of the {\it{pair empirical measures}} of the environment Markov chain. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit (variational) formulas. Our results are based on invoking ergodicity arguments in this non-elliptic set up to contr…
We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $Ω$ of boundary $Γ$ in $\mathbb {R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$ representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behavior of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviatio…
We give an elementary proof that Talagrand's sub-Gaussian concentration inequality implies a limit shape theorem for first passage percolation on any Cayley graph of Z^d, with a bound on the speed of convergence that slightly improves Alexander's bounds. Our approach, which does not use the subadditive theorem, is based on proving that the average distance is close to being geodesic. Our key observation, of independent interest, is that the problem of estimating the rate of convergence for the average distance is equivalent (in a precise sense) to estimating its "level of geodesicity".
We consider the standard first passage percolation model in $\mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $τ$ between the lower half and the upper half of the box, and the maximal flow $ϕ$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $τ$ in rational directions. Kesten and Zhang have proved the law of large numbers for $τ$ and $ϕ$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same de…
We study the 2d-Ising model defined on finite boxes at temperatures that are below but very close from the critical point. When the temperature approaches the critical point and the size of the box grows fast enough, we establish large deviations estimates on FK-percolation events that concern the phenomenon of phase coexistence.
We consider the exactly solvable model of exponential directed last passage percolation on $\mathbb{Z}^2$ in the large deviation regime. Conditional on the upper tail large deviation event $\mathcal{U}_δ:=\{T_{n}\geq (4+δ)n\}$ where $T_{n}$ denotes the last passage time from $(1,1)$ to $(n,n)$, we study the geometry of the polymer/geodesic $Γ_{n}$, i.e., the optimal path attaining $T_{n}$. We show that conditioning on $\mathcal{U}_δ$ changes the transversal fluctuation exponent from the characteristic $2/3$ of the KPZ universality class to $1/2$, i.e., conditionally, the smallest strip around …
We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and Häggström and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, namely, for some critical value $λ_{\mathrm{c}} >0$ of the bias, it holds that the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk is strictly positive if the bias $λ$ is strictly smaller than $λ_{\mathrm{c}}$, whereas $\overline{\mathrm{v}}=0$ if $λ\geq λ_{\mathrm{c}}$. We show that at the critical bias $λ= λ_{\mat…
We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelsson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk. Namely, there exists some critical value $λ_{\mathrm{c}}>0$ such that $\overline{\mathrm{v}}>0$ if $λ\in (0,λ_{\mathrm{c}})$ and $\overline{\mathrm{v}}=0$ if $λ>λ_{\mathrm{c}}$. We show that the speed $\overline{\mathrm{v}}$ is continuous in $λ$ on the interval $(0,λ_{\mathrm{c}})$ and differentiable on $(0,λ_{\mathrm{c}}/2)$. Mo…
In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction.
Benjamini, Lyons and Schramm [Random Walks and Discrete Potential Theory (1999) 56-84] considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve several problems raised by those authors. The anchored expansion constant is a variant of the Cheeger constant; its positivity implies positive lower speed for the simple random walk, as shown by Virag [Geom. Funct. Anal. 10 (2000) 1588-1605]. We prove that if G has a positive anchored expansion constant, then so does every infinite cluster of independent percolat…
The focus of this article is on the different behavior of large deviations of random subadditive functionals above the mean versus large deviations below the mean in two random media models. We consider the point-to-point first passage percolation time $a_n$ on $\mathbb{Z}^d$ and a last passage percolation time $Z_n$. For these functionals, we have $\lim_{n\to\infty}\frac{a_n}{n}=ν$ and $\lim_{n\to\infty}\frac{Z_n}{n}=μ$. Typically, the large deviations for such functionals exhibits a strong asymmetry, large deviations above the limiting value are radically different from large deviations belo…
We show that on a Cayley graph of a nonamenable group, almost surely the infinite clusters of Bernoulli percolation are transient for simple random walk, that simple random walk on these clusters has positive speed, and that these clusters admit bounded harmonic functions. A principal new finding on which these results are based is that such clusters admit invariant random subgraphs with positive isoperimetric constant. We also show that percolation clusters in any amenable Cayley graph almost surely admit no nonconstant harmonic Dirichlet functions. Conversely, on a Cayley graph admitting non…
