Disconnection and entropic repulsion for the harmonic crystal with random conductances

Abstract

We study level-set percolation for the harmonic crystal on ${\mathbb{Z}}^d$, $d\ge 3$, with uniformly elliptic random conductances. We prove that this model undergoes a non-trivial phase transition at a critical level that is almost surely constant under the environment measure. Moreover, we study the disconnection event that the level-set of this field below a level $\alpha$ disconnects the discrete blow-up of a compact set $A\subseteq {\mathbb{R}}^d$ from the boundary of an enclosing box. We obtain quenched asymptotic upper and lower bounds on its probability in terms of the homogenized capacity of $A$, utilizing results from Neukamm, Schäffner and Schlömerkemper, see arXiv:1606.06533. Furthermore, we give upper bounds on the probability that a local average of the field deviates from some profile function depending on $A$, when disconnection occurs. The upper and lower bounds concerning disconnection that we derive are plausibly matching at leading order. In this case, this work shows that conditioning on disconnection leads to an entropic push-down of the field. The results in this article generalize the findings of arXiv:1802.02518 and arXiv:1808.09947 by the authors which treat the case of constant conductances. Our proofs involve novel "solidification estimates" for random walks, which are similar in nature to the corresponding estimates for Brownian motion derived by Sznitman and the second author in arXiv:1706.07229.

arXiv comment58 pages, 2 figures, to appear in Communications in Mathematical Physics

References in corpus (29)

• Random walks on supercritical percolation clusters
• Recent progress on the Random Conductance Model
• From loop clusters and random interlacement to the free field
• Phase transition and level-set percolation for the Gaussian free field
• Soft local times and decoupling of random interlacements
• Invariance principle for the random conductance model with unbounded conductances
• On chemical distances and shape theorems in percolation models with long-range correlations
• On decoupling inequalities and percolation of excursion sets of the Gaussian free field
• Elliptic regularity and quantitative homogenization on percolation clusters
• Equality of critical parameters for percolation of Gaussian free field level-sets
• Invariance principle for the random conductance model in a degenerate ergodic environment
• Quenched invariance principle for random walks with time-dependent ergodic degenerate weights
• Scaling limit for a class of gradient fields with nonconvex potentials
• Limit theory for random walks in degenerate time-dependent random environments
• Disconnection, random walks, and random interlacements
• The sign clusters of the massless Gaussian free field percolate on ${\mathbb{Z}}^d$, $d⩾3$ (and more)
• On the radius of Gaussian free field excursion clusters
• A lower bound for disconnection by random interlacements
• On macroscopic holes in some supercritical strongly dependent percolation models
• Geometry of Gaussian free field sign clusters and random interlacements
• Stochastic homogenization of nonconvex discrete energies with degenerate growth
• Solidification of porous interfaces and disconnection
• Disconnection by level sets of the discrete Gaussian free field and entropic repulsion
• A lower bound for disconnection by simple random walk
• Excess deviations for points disconnected by random interlacements
• On bulk deviations for the local behavior of random interlacements
• Large deviations for occupation time profiles of random interlacements
• Quenched Invariance Principle for a class of random conductance models with long-range jumps
• Decoupling inequalities for the Ginzburg-Landau $\nabla \phi$ models

Cited by in corpus (4)

• On the radius of Gaussian free field excursion clusters
• On the cost of the bubble set for random interlacements
• First passage percolation with long-range correlations and applications to random Schrödinger operators
• Generating Galton-Watson trees using random walks and percolation for the Gaussian free field