Factorization in the multirefined tangent method

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 where 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 is fully computable in terms of the large deviation function L introduced in Debin et al (2019 J. Stat. Mech. 113107) (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 the 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.

Double tangent method for two-periodic Aztec diamonds

“…It is surprisingly much simpler than in the case H ′ (0; r, s) > 0, and strangely is the sum of two decoupled terms, one in r and one in s. Moreover that function F 1 (z) is exactly the one controlling the asymptotic value of the one-refined partition function (this was also observed in other models [DR21]). This observation is again fully consistent with the factorization discussed above and the specific value of Z up 1 , as the integral of L(f ′ ), see (6.1).…”

“…Moreover the uppermost path being a deterministic trajectory f in that limit, Z up 1 can be given an explicit form in terms of the function L(t) associated with the lattice paths (see Section 4.2). Altogether the proposal of [DR21] As a first remark, we observe that for the two-refined partition function, the 1-to-0 transition readily follows from the previous formula, because it predicts the following identity, (6.2) Indeed if r, s are sufficiently large so that H ′ (0; r, s) 0, the uppermost trajectory f is the blue line in the left panel of Figure 5, made of three rectilinear sections. The integral of L(f ′ ) is trivial and yields Given the form of F 2 (r, s) in (5.17) when H ′ (0; r, s) 0, the identity (6.2) is non-trivial and somewhat surprising.…”

“…The tangent method, as originally proposed in [CS16], required an extension of the domain of interest and a saddle point analysis to determine the likeliest entry point in the domain. It has been slightly reformulated in [DR21] in a way that does not require an extension of the domain but instead uses directly the one-point boundary function. We briefly review it.…”

“…The reformulation is based on a factorization property: since the uppermost path has no effect, in the scaling limit, on what the other paths do, in particular on the arctic curve itself, the partition function of the complete set of n paths should factorize, at dominant order, into that of the smaller system with n − 1 paths and the partition function Z up 1 for the single, uppermost path, constrained not to cross the arctic curve and to stay in the domain. Applied to two-periodic diamonds for which the uppermost path is conditioned to leave the westnorth boundary at the point P ′ 0 , the factorization leads to [DR21] lim…”

“…We would like to make a few general remarks on the results obtained in the last two sections. Let us recall, as explained in Section 4.3, that the factorization proposal made in [DR21] is that in the scaling limit, the ratio of partition functions Z n /Z n−1 where Z n may or may not be refined, is equal to the contribution of the uppermost path Z up 1 . Moreover the uppermost path being a deterministic trajectory f in that limit, Z up 1 can be given an explicit form in terms of the function L(t) associated with the lattice paths (see Section 4.2).…”