Fluctuations of the log-gamma polymer free energy with general parameters and slopes

Abstract

We prove that the free energy of the log-gamma polymer between lattice points (1, 1) and (M, N ) converges to the GUE Tracy-Widom distribution in the M 1/3 scaling, provided that N/M remains bounded away from zero and infinity. We prove this result for the model with inverse gamma weights of any shape parameter θ > 0 and furthermore establish a moderate deviation estimate for the upper tail of the free energy in this case. Finally, we consider a non i.i.d. setting where the weights on finitely many rows and columns have different parameters, and we show that when these parameters are critically scaled the limiting free energy fluctuations are governed by a generalization of the Baik-Ben Arous-Péché distribution from spiked random matrices with two sets of parameters. Contents Introduction and main results1 2. Laplace transform formula 8 3. Asymptotic analysis of Fredholm determinants 24 4.

Maximal free energy of the log-gamma polymer

“…Let us turn now to the higher resolution information contained in our results, beyond the limit of N −1 F N . In the subcritical regime, θ ∈ (0, θ c ), we show that F N and log Z(1, 1; N, N ) behave the same down to order N 1/3 , and thus conclude our theorem by known asymptotic results for the later random variable [BCR13, KQ18,BCD20]. In the supercritical regime, θ > θ c one might hope that the maximizing points will be order one apart, and hence the log(N ) behavior for F N makes sense from standard extreme value theory.…”

Hidden diagonal integrability of q-Hahn vertex model and Beta polymer model

“…Previously, only Beta polymer models inhomogeneous along two directions at most were known to be integrable. Moreover, to the best of our knowledge, this is the first integrable example of a directed polymer model inhomogeneous in three directions (there exist several related models with two families of parameters, studied in [5,23,42]). We expect that a lot of currently known properties of the Beta polymer can be extended to our inhomogeneous version, moreover, some new properties might emerge based on the new extension.…”

“…The color 0 is commonly reserved for the absence of a path 5. In the case n = 1 the vertex weights (2.3) reproduce the orthogonality weights for the q-Hahn polynomials, hence the name.…”

The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall

“…Remark: Note that for the same log-gamma polymer model, Fredholm determinant formulas are available in [18,58,59], and after taking the limit to the KPZ equation, this should allow to match with (78) but this route is more technical and we will not pursue it here.…”

“…These conditions can be satisfied only for small enough k. The contours may be taken as closed curves or be deformed to become infinite vertical lines. It seems that the formula (89) has not been written anywhere in the literature (though Fredholm determinant formulas for the Laplace transform are given in [18,58,59]). This moment formula can be obtained by taking appropriate specializations and limits in [61, Theorem 4.6] (the appropriate specializations and limits that one needs to take are explained in many references, see e.g.…”