Abstract
We consider a continuous time random walk on the two-dimensional discrete torus, whose motion is governed by the discrete Gaussian free field on the corresponding box acting as a potential. More precisely, at any vertex the walk waits an exponentially distributed time with mean given by the exponential of the field and then jumps to one of its neighbors, chosen uniformly at random. We prove that throughout the low-temperature regime and at in-equilibrium timescales, the process admits a scaling limit as a spatial K-process driven by a random trapping landscape, which is explicitly related to the limiting extremal process of the field. Alternatively, the limiting process is a supercritical Liouville Brownian motion with respect to the continuum Gaussian free field on the box. This demonstrates rigorously and for the first time, as far as we know, a dynamical freezing in a spin glass system with logarithmically correlated energy levels.
Highlights
1.1 Setup and main result Let VN := [0, N )2 ∩ Z2 be the discrete box of side length N and consider the random field hN ≡x∈VN having the law of the discrete Gaussian free field (DGFF) on VN with zero boundary conditions
At in-equilibrium timescales, large N and low temperatures, the dynamics are effectively governed by the extreme values of the underlying Gaussian free field, which determine the trapping landscape
We first use the results from the previous section to prove that the trace process can be coupled together with the χ-driven spatial pre K-process, so that with high probability, they are close in the
Summary
1.1 Setup and main result Let VN := [0, N )2 ∩ Z2 be the discrete box of side length N and consider the random field hN ≡ (hN,x)x∈VN having the law of the discrete Gaussian free field (DGFF) on VN with zero boundary conditions. On the same probability space, define a process XN ≡ (XN (t) : t ≥ 0) taking values in the two-dimensional discrete torus VN∗ = Z2/N Z2, whose vertices we identify with. PPP stands for Poisson point process, and the law of η should be interpreted as given conditionally on Z, which is assumed to be defined on the same probability space as η itself. The second ingredient in the construction of the limiting object is a point process which we denote, for β > α, by χ ≡ χ(β). It is defined on the same probability space as the measure Z and its law is given conditionally on Z as χ(β) ∼ PPP Z(dz) ⊗ κβ|Z|t−1−α/βdt ,.
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