P-adic numbers and kernels

TL;DR

This paper explores the connection between p-adic numbers and kernels, applying this framework to spin glasses, number theory, and statistical models, revealing new interpretations and links between these mathematical areas.

Contribution

It introduces a kernel-based perspective on p-adic numbers, linking number theory, statistical physics, and large deviation theory with novel interpretations of models like Derrida's GREM.

Findings

Derrida's GREM can be viewed as a random numerical base.

Established a kernel representation of the Primon gas.

Linked p-adic number properties to kernel theory and number theory.

Abstract

We discuss the relation between p-adic numbers and kernels in view of a recent large deviation theory for mean-field spin glasses. As an application we show several fundamental properties of numerical bases in kernel language. In particular, we show that the Derrida's Generalized Random Energy Model can be interpreted as a (random) numerical base. We also show an application to the Primon gas and the Riemann Zeta Function by constructing a kernel representation of the Primon gas based on a finite p-base, thereby establishing a concrete link between number theory and kernel theory.