Factorization in the multirefined tangent method

TL;DR

This paper establishes a factorization property in the tangent method for statistical systems with arctic curves, enabling explicit computation of refined partition functions and simplifying the derivation of arctic curves.

Contribution

It proves a factorization property of refined partition functions, connecting external path contributions to large deviation functions, and reformulates the tangent method without domain extension.

Findings

Verified factorization in Aztec diamond and alternating sign matrices

Derived explicit formulas for asymptotics of refined partition functions

Reformulated tangent method to reveal the role of the large deviation function

Abstract

When applied to statistical systems showing an arctic curve phenomenon, the tangent method assumes that a modification of the most external path does not affect the arctic curve. We strengthen this statement and also make it more concrete by observing a factorization property: if $Z^{}_{n+k}$ denotes a refined partition function of a system of $n+k$ non-crossing paths, with the endpoints of the $k$ most external paths possibly displaced, then at dominant order in $n$, it factorizes as $Z^{}_{n+k} \simeq Z^{}_{n} Z_k^{\rm out}$ where $Z_k^{\rm out}$ is the contribution of the $k$ most external paths. Moreover if the shape of the arctic curve is known, we find that the asymptotic value of $Z_k^{\rm out}$ is fully computable in terms of the large deviation function $L$ introduced in \cite{DGR19} (also called Lagrangean function). We present detailed verifications of the factorization in the Aztec diamond and for alternating sign matrices by using exact lattice results. Reversing the argument, we reformulate the tangent method in a way that no longer requires an extension of the domain, and which reveals the hidden role of the $L$ function. As a by-product, the factorization property provides an efficient way to conjecture the asymptotics of multirefined partition functions.