اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدExponential growth of ponds in invasion percolation on regular trees
163
0
0.0
(
0
)
تأليف
Jesse Goodman
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
We study level-set percolation of the Gaussian free field on the infinite $d$-regular tree for fixed $dgeq 3$. Denoting by $h_star$ the critical value, we obtain the following results: for $h>h_star$ we derive estimates on conditional exponential mom
ents of the size of a fixed connected component of the level set above level $h$; for $h<h_star$ we prove that the number of vertices connected over distance $k$ above level $h$ to a fixed vertex grows exponentially in $k$ with positive probability. Furthermore, we show that the percolation probability is a continuous function of the level $h$, at least away from the critical value $h_star$. Along the way we also obtain matching upper and lower bounds on the eigenfunctions involved in the spectral characterisation of the critical value $h_star$ and link the probability of a non-vanishing limit of the martingale used therein to the percolation probability. A number of the results derived here are applied in the accompanying paper [AC2].
We study the supercritical contact process on Galton-Watson trees and periodic trees. We prove that if the contact process survives weakly then it dominates a supercritical Crump-Mode-Jagers branching process. Hence the number of infected sites grows
exponentially fast. As a consequence we conclude that the contact process dies out at the critical value $lambda_1$ for weak survival, and the survival probability $p(lambda)$ is continuous with respect to the infection rate $lambda$. Applying this fact, we show the contact process on a general periodic tree experiences two phase transitions in the sense that $lambda_1<lambda_2$, which confirms a conjecture of Staceys cite{Stacey}. We also prove that if the contact process survives strongly at $lambda$ then it survives strongly at a $lambda<lambda$, which implies that the process does not survive strongly at the critical value $lambda_2$ for strong survival.
We consider invasion percolation on the square lattice. It has been proved by van den Berg, Peres, Sidoravicius and Vares, that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n from the pro
bability that in critical Bernoulli percolation the radius of an open cluster is larger than n. We show that these two probabilities are, in fact, of the same order. Moreover, we prove an analogous result for the volume of a pond.
In this paper we study a variation of the accessibility percolation model, this is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability of having
a monotone subsequence of a path from the root to a leaf, where any $k$ consecutive vertices in the path contain at least one vertex of the subsequence. An $n$-ary tree, with height $h$, is a tree whose vertices at distance at most $h-1$ to the root have $n$ children. For the case of $n$-ary trees, we prove that, as $h$ tends to infinity the probability of having such subsequence: tends to 1, if $n$ grows significantly faster than $sqrt[k]{h/(ek)}$; and tends to 0, if $n$ grows significantly slower than $sqrt[k]{h/(ek)}$.
We consider a class of reinforcement processes, called WARMs, on tree graphs. These processes involve a parameter $alpha$ which governs the strength of the reinforcement, and a collection of Poisson processes indexed by the vertices of the graph. It
has recently been proved that for any fixed bounded degree graph with Poisson firing rates that are uniformly bounded above, in the very strong reinforcement regime ($alphagg 1$ sufficiently large depending on the maximal degree), the set of edges that survive (i.e. that are reinforced infinitely often by the process) has only finite connected components. The present paper is devoted to the construction of an example in the opposite direction, that is, with the set of surviving edges having infinite connected components. Namely, we show that for each fixed $alpha>1$ one can find a regular rooted tree and firing rates that are uniformly bounded from above, for which there are infinite components almost surely. Joining such examples, we find a graph (with unbounded degrees) on which for any $alpha>1$ almost surely there are infinite connected components of surviving edges.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
