اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدNon-equilibrium Fluctuations of the Weakly Asymmetric Normalized Binary Contact Path Process
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This paper is a further investigation of the problem studied in cite{xue2020hydrodynamics}, where the authors proved a law of large numbers for the empirical measure of the weakly asymmetric normalized binary contact path process on $mathbb{Z}^d,, d geq 3$, and then conjectured that a central limit theorem should hold under a non-equilibrium initial condition. We prove that the aforesaid conjecture is true when the dimension $d$ of the underlying lattice and the infection rate $lambda$ of the process are sufficiently large.
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In this paper we are concerned with the binary contact path process introduced in cite{Gri1983} on the lattice $mathbb{Z}^d$ with $dgeq 3$. Our main result gives a hydrodynamic limit of the process, which is the solution to a heat equation. The proof
of our result follows the strategy introduced in cite{kipnis+landim99} to give hydrodynamic limit of the SEP model with some details modified since the states of all vertices are not uniformly bounded for the binary contact path process. In the modifications, the theory of the linear system introduced in cite{Lig1985} is utilized.
The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
We consider a contact process on $Z^d$ with two species that interact in a symbiotic manner. Each site can either be vacant or occupied by individuals of species $A$ and/or $B$. Multiple occupancy by the same species at a single site is prohibited. T
he name symbiotic comes from the fact that if only one species is present at a site then that particle dies with rate 1 but if both species are present then the death rate is reduced to $mu le 1$ for each particle at that site. We show the critical birth rate $lambda_c(mu)$ for weak survival is of order $sqrt{mu}$ as $mu to 0$. Mean-field calculations predict that when $mu < 1/2$ there is a discontinuous transition as $lambda$ is varied. In contrast, we show that, in any dimension, the phase transition is continuous. To be fair to physicists the paper that introduced the model, the authors say that the symbiotic contact process is in the directed percolation universality class and hence has a continuous transition. However, a 2018 paper asserts that the transition is discontinuous above the upper critical dimension, which is 4 for oriented percolation.
A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that on homogeneous trees the critical values $lambda_1$ and $lambda_2$ for global and local survival were different. He also considered trees with pe
riodic degree sequences, and Galton-Watson trees. Here, we will consider periodic trees in which the number of children in successive generation is $(n,a_1,ldots, a_k)$ with $max_i a_i le Cn^{1-delta}$ and $log(a_1 cdots a_k)/log n to b$ as $ntoinfty$. We show that the critical value for local survival is asymptotically $sqrt{c (log n)/n}$ where $c=(k-b)/2$. This supports Pemantles claim that the critical value is largely determined by the maximum degree, but it also shows that the smaller degrees can make a significant contribution to the answer.
A little over 25 years ago Pemantle pioneered the study of the contact process on trees, and showed that the critical values $lambda_1$ and $lambda_2$ for global and local survival were different. Here, we will consider the case of trees in which the
degrees of vertices are periodic. We will compute bounds on $lambda_1$ and $lambda_2$ and for the corresponding critical values $lambda_g$ and $lambda_ell$ for branching random walk. Much of what we find for period two $(a,b)$ trees was known to Pemantle. However, two significant new results give sharp asymptotics for the critical value $lambda_2$ of $(1,n)$ trees and generalize that result to the $(a_1,ldots, a_k, n)$ tree when $max_i a_i le n^{1-epsilon}$ and $a_1 cdots a_k = n^b$. We also give results for $lambda_g$ and $lambda_ell$ on $(a,b,c)$ trees. Since the values come from solving cubic equations, the explicit formulas are not pretty, but it is surprising that they depend only on $a+b+c$ and $abc$.
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