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We study random field Ising model on Z 2 where the external field is given by i.i.d. Gaussian variables with mean zero and positive variance. We show that at zero temperature the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary.

We study the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. In general, we bound the limit from above by the unique solution to a certain Hamilton-Jacobi equation. Under additional assumptions on the nonlinearity in the equation which is determined explicitly by the model, we identify the limit with the solution. Two notions of solutions, weak solutions and viscosity solutions, are considered, each of which has its own advantages and requires different treatments. For concreteness, we apply our results to a model with i.i.d. entries and symmetric interactions. In particular, for the first order and even order tensor products, we identify the limit and obtain estimates on convergence rates; for other odd orders, upper bounds are obtained.

We study random field Ising model on Z 2 where the external field is given by i.i.d. Gaussian variables with mean zero and positive variance. We show that at any positive temperature the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary. * Partially supported by NSF grant DMS-1757479 and an Alfred Sloan fellowship. Intrinsic distance on disagreements via a perturbation argumentFor any A ⊂ Z 2 , we denote by d A (•, •) the intrinsic distance on A, i.e., the graph distance on the induced subgraph on A. Let σ Λ N ,± be spins sampled according to µ Λ N ,± . We will use repeatedly the standard monotonicity properties of the Ising model with respect to external fields and boundary conditions (c.f. [2, Section 2.2] for detailed discussions). Let π be a monotone coupling of µ Λ N ,±