Large Deviations for $\nabla_{\varphi}$ Interface Model and Derivation of Free Boundary Problems

Concentration under scaling limits for weakly pinned Gaussian random walks

“…We may assume the condition (4.8) for g ∈ H 1 a,b (D), cf. [12]; if K = 0 (i.e., g(t) = 0 for all t ∈ D), (4.6) follows from Proposition 4.2 and (4.9). Determine j N from t in (4.8) by j p,N = [N t p ], 1 ≤ p ≤ K , = 1, 2, which is macroscopically t.…”

“…Indeed, one can give the proof of Theorem 4.1 essentially just by copying the proof stated in [12] line by line. But, for completeness, we give another proof with slightly different flavor, which might be simpler in some aspect.…”

“…The LDP for µ D,ε N is shown in [12], Theorem 2.2, when d = 1. Indeed, one can give the proof of Theorem 4.1 essentially just by copying the proof stated in [12] line by line.…”

“…Proof We first discuss the situation without a wall. The assertion for µ a,F N follows by Schilder's theorem (or Mogul'skii's theorem [16], [4]), while for µ a,b N , we may employ the contraction principle for the LDP in addition as in the proof of Lemma 6.1 of [12].…”

“…4 (or Theorem 2.2 of [12] for µ D N when d = 1 and m = 0, and [4,16] when ε = 0). The speeds are always N and the unnormalized rate functionals are given by D , D,+ , F and F,+ , respectively, all of which are of the form: (1) = b} in the Dirichlet case respectively h ∈ H 1 a,F (D) = {h ∈ H 1 (D); h(0) = a} in the free case with certain non-negative constants ξ , where | · | stands for the Lebesgue measure on D and…”