## Tohoku Mathematical Journal

### Infinite particle systems of long range jumps with long range interactions

Syota Esaki

#### Abstract

In this paper a general theorem for constructing infinite particle systems of jump type 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 71, Number 1 (2019), 9-33.

Dates
First available in Project Euclid: 9 March 2019

https://projecteuclid.org/euclid.tmj/1552100440

Digital Object Identifier
doi:10.2748/tmj/1552100440

Mathematical Reviews number (MathSciNet)
MR3920788

Zentralblatt MATH identifier
07060324

#### Citation

Esaki, Syota. Infinite particle systems of long range jumps with long range interactions. Tohoku Math. J. (2) 71 (2019), no. 1, 9--33. doi:10.2748/tmj/1552100440. https://projecteuclid.org/euclid.tmj/1552100440

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