Correlation decay in area-tilted line ensembles

Abstract

Random surfaces on a hard substrate often exhibit entropic repulsion, wherein the surface is propelled upwards to allow entropically preferable downward fluctuations. A particularly rich class of examples arises from the low-temperature 3D Ising model. A powerful approach to studying such surfaces is through their level curves, which form a family of non-intersecting random curves. In [CIW18, CIW19], an ensemble of Brownian lines with geometrically increasing area tilts was proposed as a putative limiting model in this case. This model falls outside the scope of techniques based on integrable or SDE structures, which have been key ingredients in the study of the Airy line ensemble. A particularly intriguing question about such line ensembles concerns their mixing properties when viewed as a Markov process, and in particular the rate of decay of correlations in time. For the Airy line ensemble, this decay is known to be inverse quadratic. The first quantitative bound on the decay of correlations in the area-tilted model, established in [CG25], was slower than polynomial in time. An earlier result [DLZ24] had established positivity of the spectral gap for the finite-line version of the ensemble, without quantitative bounds. This left open the important question of the true decay rate of correlations for the infinite ensemble. Settling this question for sufficiently large area-tilt strength, corresponding to sufficiently low temperature for the 3D Ising model, we prove exponential decay of correlations for the infinite ensemble and a uniform (in the number of lines) positive spectral gap for the finite ensemble. Our proof is based on establishing a precise form of separation of scales between curves of different indices, using a novel probabilistic approach involving embedding supercritical branching processes in the line ensemble.