Positive Discrete Spectrum of the Evolutionary Operator of Supercritical Branching Walks with Heavy Tails

TL;DR

This paper investigates the spectral properties of the evolutionary operator in supercritical branching random walks with heavy-tailed jumps, revealing bounds on positive eigenvalues and conditions for multiple eigenvalues due to spatial symmetry.

Contribution

It provides new insights into the spectrum of the evolutionary operator for supercritical branching walks with heavy tails, including bounds on positive eigenvalues and symmetry-induced multiple eigenvalues.

Findings

Number of positive eigenvalues does not exceed the number of sources.

Maximal eigenvalue is always simple.

Symmetry in source configuration can cause multiple eigenvalues.

Abstract

We consider a continuous-time symmetric supercritical branching random walk on a multidimensional lattice with a finite set of the particle generation centres, i.e. branching sources. The main object of study is the evolutionary operator for the mean number of particles both at an arbitrary point and on the entire lattice. The existence of positive eigenvalues in the spectrum of an evolutionary operator results in an exponential growth of the number of particles in branching random walks, called supercritical in the such case. For supercritical branching random walks, it is shown that the amount of positive eigenvalues of the evolutionary operator, counting their multiplicity, does not exceed the amount of branching sources on the lattice, while the maximal of these eigenvalues is always simple. We demonstrate that the appearance of multiple lower eigenvalues in the spectrum of the evolutionary operator can be caused by a kind of `symmetry' in the spatial configuration of branching sources. The presented results are based on Green's function representation of transition probabilities of an underlying random walk and cover not only the case of the finite variance of jumps but also a less studied case of infinite variance of jumps.