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

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 $\ast$-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.

arXiv comment9 pages, 1 figure; to appear in Electronic Communications in Probability