Infinite particle systems of long range jumps with long range interactions

TL;DR

This paper develops a general framework for constructing infinite particle systems with long-range jumps and interactions, applicable to systems with stable processes and logarithmic or polynomial decay potentials, expanding understanding of such complex systems.

Contribution

It introduces a broad theorem for constructing infinite particle systems with long-range interactions, applicable to various potentials and stable processes, under certain conditions.

Findings

Construction possible for any α in (0, 2) with translation-invariant equilibrium measure.

System can be constructed for potentials from random matrix theory and Ruelle's class.

Restrictions depend on the growth of the 1-correlation function of the measure.

Abstract

In this paper a general theorem of constructing infinite particle systems of jump types with long range interactions is presented. It can be applied to the system that each particle undergoes an $\alpha$-stable process and interaction between particles is given by the logarithmic potential appearing random matrix theory or potentials of Ruelle's class with polynomial decay. It is shown that the system can be constructed for any $\alpha \in (0, 2)$ if its equilibrium measure $\mu$ is translation invariant, and $\alpha$ is restricted by the growth order of the 1-correlation function of the measure $\mu$ in general case.