On the exponential growth rates of lattice animals and interfaces II:\n new asymptotic bounds

2021/07/13 by Agelos Georgakopoulos, Christoforos Panagiotis, Georgakopoulos, Agelos +1
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2107.06272

Abstract

We introduce a method for translating any upper bound on the percolation\nthreshold of a lattice G into a lower bound on the exponential growth rate\na(G) of lattice animals and vice-versa. We exploit this in both directions.\nWe improve on the best known asymptotic lower and upper bounds on\na(\ℤ^d) as d\→ \∞. We use percolation as a tool to obtain the\nlatter, and conversely we use the former to obtain lower bounds on\np_c(\ℤ^d). We obtain the rigorous lower bound\n\p_c(\ℤ³)>0.2522 for 3-dimensional site percolation.\n

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