Matrix continued fractions associated with lattice paths, resolvents of difference operators, and random polynomials

Abstract

We begin our analysis with the study of two collections of lattice paths in the plane, denoted $\mathcal{D}_{[n,i,j]}$ and $\mathcal{P}_{[n,i,j]}$. These paths consist of sequences of $n$ steps, where each step allows movement in three directions: upward (with a maximum displacement of $q$ units), rightward (exactly one unit), or downward (with a maximum displacement of $p$ units). The paths start from the point $(0,i)$ and end at the point $(n,j)$. In the collection $\mathcal{D}_{[n,i,j]}$, it is a crucial constraint that paths never go below the $x$-axis, while in the collection $\mathcal{P}_{[n,i,j]}$, paths have no such restriction. We assign weights to each path in both collections and introduce weight polynomials and generating series for them. Our main results demonstrate that certain matrices of size $q\times p$ associated with these generating series can be expressed as matrix continued fractions. These results extend the notable contributions previously made by P. Flajolet and G. Viennot in the scalar case $p=q=1$. The generating series can also be interpreted as resolvents of one-sided or two-sided difference operators of finite order. Additionally, we analyze a class of random banded matrices $H$, which have $p+q+1$ diagonals with entries that are independent and bounded random variables. These random variables have identical distributions along diagonals. We investigate the asymptotic behavior of the expected values of eigenvalue moments for the principal $n\times n$ truncation of $H$ as $n$ tends to infinity.