Weakly Supercritical Branching Random Walks

Last modified: 2014-04-07

#### Abstract

We consider a continuous-time simple symmetric branching random walk on multidimensional lattices under the condition that at the initial time there is only one particle on the lattice. Such models have different applications. For instance, M.Cranston et al. (2009) considered a model of a continuous homopolymer in a potential field with the Brownian motion instead of a random walk. In this work it is proved, in particilar, the existence of a critical value of temperature such that when the temperature trespasses this value the transition between globular and extended phase occurs. The approach proposed by M.Cranston et al. (2009) is based on the resolvent analysis of the evolution operator but does not take into account the case of a simple symmetric branching random walk on multidimensional lattices. In our work, the underlying random walk is determined by the discrete Laplace operator on the multidimensional lattice. In this case the related Green's function coincides with the Laplace transform of the transition probability of random walk. The process of branching in the source situated in one point of the lattice is defined by the infinitesimal generating function. One of important characteristics of the process is the parameter beta defined as the value of the first derivative of this function at the point 1. The offsprings of the initial particle walk, die and reproduce themselves under the same law independently from each other. As is known (see, Molchanov and Yarovaya, 2012), in this case the evolution operator of mean number of the particles is the discrete Laplace operator with one-point perturbation, and the asymptotic behaviour of the process is dependent on the intensity of the source, and therefore the value of beta (see, Bogachev and Yarovaya, 1998, Yarovaya, 2007). It is found that, there exists the critical value beta_cr such that when beta > beta_cr the evolution operator has a positive isolated eigenvalue lambda given by the values of beta and Green's function. This eigenvalue defines the exponential rate of growth of the number of particles (see, Yarovaya, 2010). The main goal of our work is to analyse the asymptotic behaviour of the eigenvalue lambda when beta converges to beta_cr. The behaviour of the branching random walk with beta converging to beta_cr is offered to call the weakly supercritical. The key moment in this problem is the analysis of the properties of Green's function with small lambda (Molchanov and Yarovaya, 2012). Based on the investigation of the properties of the Green's function we classify the asymptotic behaviour of lambda when beta converges to beta_cr with respect to dimension of the space of walk. In conclusion we should mention the actuality of the solution of this problem for a finite set of sources of branching on the lattice.

This work was partly supported by RFBR grant 13-01-00653.

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This work was partly supported by RFBR grant 13-01-00653.

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