Disconnection and Entropic Repulsion for the Harmonic Crystal with Random Conductances

Abstract

We study level-set percolation for the harmonic crystal on Z d , d ≥ 3, with uniformly elliptic random conductances. We prove that this model undergoes a nontrivial 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 α disconnects the discrete blow-up of a compact set A ⊆ 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 (SIAM J Math Anal 49(3):1761-1809, 2017). 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 Nitzschner (Electron J Probab 23:105, 2018) and Chiarini and Nitzschner (Probab Theory Relat Fields 177(1-2):525-575, 2020) 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 Nitzschner and Sznitman (

First passage percolation with long-range correlations and applications to random Schrödinger operators

“…(ii) The advantage of considering general symmetric weights (ā e ) e∈E d under condition (4.1) instead of unit weights in Proposition 4.1 is that it allows us to prove results similar to Theorems 1.3 and 1.4 for the Gaussian free field with random conductances, as studied for instance in [24]. Indeed, assume that the conductances (ā x,y ) x,y∈Z d are chosen at random under some probability Q under which they are stationary and ergodic with respect to shifts and a.s. satisfy (4.1) for some nonrandom constant C G .…”

Effect of microcarbon particle size and dispersion on the electrical conductivity of LLDPE-carbon composite

“…In a volume having a significant difference in particle sizes, the conductive ball will occupy the empty space between the larger non-conductive balls. The conductive ball is referred to as filler that fills the matrix gaps of non-conductive balls (Cipriani & van Ginkel, 2020;Chiarini & Nitzschner, 2021).…”

“…The manual sieving process is limited to controlling the mesh size by allowing many particles smaller than the mesh scale to pass through. If the lattice position during the composite solidification process is formed through the formation of nuclei, then the position is randomly occupied by filler particles 𝑚𝑚a, 𝑚𝑚b, 𝑚𝑚c, …., and so on, either as single particles or particle clusters (Chiarini & Nitzschner, 2021). Particle clusters may occur due to the interaction between fillers in the composite, such as van der Waals bonding.…”