Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix

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.