Explicit finite size corrections for Toeplitz determinants and rectangular domino tilings

Abstract

Finite Toeplitz truncation does not, in general, preserve multiplication. For products of $m$ $N\times N$ tridiagonal Toeplitz matrices with symbols $z+z^{-1}+a_h$, we make the two boundary corrections in Widom's product formula for finite sections explicit. The method of images gives two Hankel corrections when $m\le N+2$, a sharp uniform degree bound. For $N\ge m-1$, the ratio of the determinant of the matrix product to the Toeplitz determinant with the product symbol reduces, whenever defined, to a determinant of order $2(m-1)$, which reversal symmetry splits into two determinants of order $m-1$. A rectangular Schur polynomial gives an exact formula for the Toeplitz determinant with the product symbol, including when roots coincide. For symbols nonzero on the unit circle, classical identities for rational symbols give an exact $m\times m$ determinant formula for the ratio, with an explicit confluent form for repeated roots. With $ρ_h$ the zero of $z+z^{-1}+a_h$ in $|z|<1$, the limit is $\prod_{i<j}(1-ρ_iρ_j)^2$ as $N\to\infty$ at fixed $m,a_h$. For rectangular domino tilings, these formulas compare the classical Kasteleyn--Temperley--Fisher product with the banded Toeplitz determinant formed from the product symbol. We obtain the first exponentially small correction at fixed width and relate this comparison to the Toeplitz determinant furnished by the dual Cauchy identity.