Lozenge tilings and the Gaussian free field on a cylinder

Abstract

We use the periodic Schur process, introduced in [Bor07], to study the random height function of lozenge tilings (equivalently, dimers) on an infinite cylinder distributed under two variants of the q vol measure. Under the first variant, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under the second variant, corresponding to an unrestricted dimer model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured in [Gor21] for dimer models on planar domains with holes. Contents1. Introduction 1.1. Background 1.2. Lozenge tilings and cylindric partitions 1.3. Main results 1.4. Methods 1.5. Organization. 2. Gaussian free field and Green's function on cylinder 2.1. Gaussian free field 2.2. Green's function 3. The model 3.1. Periodic Schur processes 3.2. Lozenge tilings 3.3. q vol Measure on cylindric partitions 4. Moment formula 5. Asymptotics 6. Limit shape and fluctuations 6.1. Limit shape 6.2. Gaussian free field convergence for the unshifted model 6.3. Gaussian free field convergence for the shift-mixed model 7. Matching with conjectures on height fluctuations 7.1. Kenyon-Okounkov conjecture 7.2. Hole height fluctuations for multiply connected domains References

Identity between restricted Cauchy sums for the $q$-Whittaker and skew Schur polynomials

“…This observation has led us to a careful comparison between the structure of the q-Whittaker measure and the (shift-mixed) periodic Schur measure [Bor07], as the latter is a canonical model of free fermions at positive temperature in one dimension [BB19]. In recent years the same model has received some attention and its properties and generalizations have been considered in [BBNV18,Kos21,ARVP21]. The key observation we make is that the Fredholm determinant appearing in the shift-mixed periodic Schur measure is almost the same as the one found in [IS19] for the q-Whittaker measure, the only difference being in the contours of the contour integral expression for the kernels.…”

Peaks of cylindric plane partitions

“…Remark 8. Let us notice one last thing: for certain parameter ranges, our main theorem combined with recent results of [1] could perhaps elucidate certain (perhaps expected) behavior of maxima of Gaussian free fields on a cylinder. 2 Last passage percolation and q-Whittaker measures…”