Factorization in the multirefined tangent method

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