A note on $\alpha$-permanent and loop soup

TL;DR

This paper explores the connection between the algebraic concept of α-permanent and probabilistic loop soups, providing explicit formulas and proofs through combinatorial and probabilistic methods.

Contribution

It establishes a novel link between α-permanents and loop soups, with explicit expansions for block matrices from special matrix classes.

Findings

Explicit expansions of α-permanents for block matrices from *-forests.

Two independent proofs: combinatorial and probabilistic.

Demonstrates the relationship between algebraic permanents and probabilistic models.

Abstract

In this paper, it is shown that $\alpha$-permanent in algebra is closely related to loop soup in probability. We give explicit expansions of $\alpha$-permanents of the block matrices obtained from matrices associated to $*$-forests, which are a special class of matrices containing tridiagonal matrices. It is proved in two ways, one is the direct combinatorial proof, and the other is the probabilistic proof via loop soup.