Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix

TL;DR

This paper derives exact asymptotics for a determinant related to Pascal matrices, revealing insights into cyclically symmetric plane partitions, vicious walkers, and loop models, with applications to percolation probabilities.

Contribution

It provides the first exact asymptotic formulas for the determinant of a Pascal matrix with a complex exponential, connecting combinatorics, statistical mechanics, and probability.

Findings

Exact asymptotics of the determinant up to order L^{-14}

Explicit formulas for loop model observables

Asymptotic probabilities for percolation clusters

Abstract

We have obtained the exact asymptotics of the determinant $\det_{1\leq r,s\leq L}[\binom{r+s-2}{r-1}+\exp(i\theta)\delta_{r,s}]$. Inverse symbolic computing methods were used to obtain exact analytical expressions for all terms up to relative order $L^{-14}$ to the leading term. This determinant is known to give weighted enumerations of cyclically symmetric plane partitions, weighted enumerations of certain families of vicious walkers and it has been conjectured to be proportional to the one point function of the O$(1)$ loop model on a cylinder of circumference $L$. We apply our result to the loop model and give exact expressions for the asymptotics of the average of the number of loops surrounding a point and the fluctuation in this number. For the related bond percolation model, we give exact expressions for the asymptotics of the probability that a point is on a cluster that wraps around a cylinder of even circumference and the probability that a point is on a cluster spanning a cylinder of odd circumference.