The discrete analogue of the Gaussian

TL;DR

This paper explores the heat kernel on integers as a discrete analogue of the Gaussian, reviewing its applications and proving new theorems related to its properties and potential uses in privacy and physics.

Contribution

It introduces a new local limit theorem for integer sums and discusses novel spectral zeta function values, expanding understanding of the discrete heat kernel's applications.

Findings

Proves a new local limit theorem for sums of integer-valued variables.

Derives novel special values of the spectral zeta function of Bethe lattices.

Discusses potential applications in differential privacy.

Abstract

This paper illustrates the utility of the heat kernel on $\mathbb{Z}$ as the discrete analogue of the Gaussian density function. It is the two-variable function $K_{\mathbb{Z}}(t,x)=e^{-2t}I_{x}(2t)$ involving a Bessel function and variables $x\in\mathbb{Z}$ and real $t\geq 0$. Like its classic counterpart it appears in many mathematical and physical contexts and has a wealth of applications. Some of these will be reviewed here, concerning Bessel integrals, trigonometric sums, hypergeometric functions and asymptotics of discrete models appearing in statistical and quantum physics. Moreover, we prove a new local limit theorem for sums of integer-valued random variables, obtain novel special values of the spectral zeta function of Bethe lattices, and provide a discussion on how $e^{-2t}I_{x}(2t)$ could be useful in differential privacy.