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Register a new userAn improved upper bound for critical value of the contact process on $mathbb{Z}^d$ with $dgeq 3$
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Xiaofeng Xue
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In this paper we give an improved upper bound for critical value $lambda_c$ of the basic contact process on the lattice $mathbb{Z}^d$ with $dgeq 3$. As a direct corollary of out result, [ lambda_cleq 0.384. ] when $d=3$.
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We consider the Stavskayas process, which is a two-states Probabilistic Celular Automata defined on a one-dimensional lattice. The process is defined in such a way that the state of any vertex depends only on itself and on the state of its right-adjacent neighbor. This process was one of the first multicomponent systems with local interaction, for which has been proved rigorously the existence of a kind of phase transition. However, the exact localization of its critical value remains as an open problem. In this work we provide a new lower bound for the critical value. The last one was obtained by Andrei Toom, fifty years ago.
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In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${{mathbb{Z}}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster ${{mathcal{C}}_{infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We prove that local survival of the branching random walk on ${{mathbb{Z}}^d}$ also implies that for $k$ sufficiently large the associated $k$-type contact process survives on ${{mathcal{C}}_{infty}}$. This implies that the strong critical parameters of the branching random walk on ${{mathbb{Z}}^d}$ and on ${{mathcal{C}}_{infty}}$ coincide and that their common value is the limit of the sequence of strong critical parameters of the associated $k$-type contact processes. These results are extended to a family of restrained branching random walks, that is, branching random walks where the success of the reproduction trials decreases with the size of the population in the target site.
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Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.
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