An explicit example for the high temperature convolution: crossover between the binomial law $B(2,1/2)$ and the arcsine law

TL;DR

This paper analytically explores the high-temperature convolution of symmetric Bernoulli distributions, deriving explicit formulas that interpolate between binomial and arcsine laws, revealing a new family of probability densities.

Contribution

It provides the first explicit analytical expressions for the high-temperature convolution between two symmetric Bernoulli distributions, bridging binomial and arcsine laws.

Findings

Derived the Stieltjes transform and density for the convolution

Established a new family of densities interpolating between binomial and arcsine laws

First non-trivial explicit expression for this convolution

Abstract

In this note, we study the high-temperature convolution introduced in Ref.\ \cite{mergny_cconv}, between two symmetric Bernoulli distributions. We give an analytical expression for both the Stieltjes transform and the density. This result provides the first non-trivial expression for the high-temperature convolution of two distributions and gives a new family of densities, interpolating between the centered binomial distribution with number of trials $n{=}2$ and probability of success $p{=}1/2$, and the centered and re-scaled arcsine law.