arXiv subjects

Maximilian Nitzschner

Publications and source records attributed to Maximilian Nitzschner.

15 recordsLinked to original sources

arXiv2026

Phase transition for strongly correlated percolation models on supercritical Bernoulli clusters

We consider the level sets of the Gaussian free field and the vacant set of random interlacements, both defined on a typical realization of the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove that in the entire supercritical regime of Bernoulli bond percolation, both the level sets of the Gaussian free field and the vacant set of random interlacements undergo non-trivial percolation phase transitions at deterministic critical levels. A key aspect of the proof is the development of certain quenched controls over tree embeddings, permitting the application of a static renormalization scheme in the presence of spatial irregularities, which may be of independent interest.

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arXiv2026

On the phase transition for the number of collisions on comb graphs

We consider collisions of simple random walks on comb graphs $\mathrm{Comb}(\mathbb{Z},H)$, which are obtained by attaching vertical segments of the form $[0,H_x] \cap \mathbb{Z}$ to any point $x$ of the integer axis. For $\mathrm{Comb}(\mathbb{Z},H)$ with profile $H_x(x) = |x| \log^\gamma(|x| \vee 1)$, we show that two independent simple random walks starting from the same site collide infinitely often almost surely if $\gamma \leq 2$. If the tooth profile is taken as a typical realization of i.i.d. heavy-tailed random variables with $\textbf{P}(H_x > z) \sim Cz^{-\gamma}$ (with some $C > 0$) as $z$ tends to infinity, we show that infinitely many collisions occur almost surely for two independent random walks if $\gamma > 1/3$, whereas finitely many collisions occur almost surely if $\gamma \in (0,1/3)$, and for any $\gamma \in (0,1]$, three independent random walks only collide finitely many times, almost surely.

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arXiv2025

Non-coincidence of critical points for directed polymers on supercritical percolation clusters

We consider the model of a directed polymer in a random environment defined on the infinite cluster of supercritical Bernoulli bond percolation in dimensions $d \geq 3$. For this model, it was proved in arXiv:2205.06206 that for almost every realization of the cluster, the polymer is in a strong disorder regime for any positive inverse temperature. Here, we show for almost every realization of the cluster the existence of a non-empty sub-phase of the strong disorder regime, consisting of positive inverse temperatures in which very strong disorder does not hold. This is in contrast to the recently established sharpness of the phase transition for the directed polymer on the full lattice, see arXiv:2402.02562, arXiv:2502.04113.

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arXiv2025

Collisions of random walks on comb graphs with a planar base

In this article we study collisions of two independent random walks on comb graphs $\mathrm{Comb}(\tilde{G},f)$ for a large class of recurrent planar graphs $\tilde{G}$ and profile functions $f$, the latter governing the length of vertical segments (called "teeth") attached to vertices of the base graph $\tilde{G}$. We prove that the number of collisions of two random walks starting from the same site undergoes a phase transition depending on the growth of $f$. As a benchmark example, we show that for $\mathrm{Comb}(\mathbb{Z}^2,f_\gamma)$ with $f_\gamma(z) = \log^{\gamma}(\|z\|_\infty\vee 1)$ and $\|\cdot \|_\infty$ denoting the supremum norm, two independent random walks started at the origin collide finitely often almost surely if $\gamma > 1$, answering a question of Barlow, Peres, and Sousi, see arXiv:1003.3255, who established that infinitely many collisions occur almost surely if $\gamma \leq 1$. We furthermore establish phase transitions in the cases where the base graph $\tilde{G}$ is pre-fractal, or a typical realization of a supercritical cluster of planar Bernoulli bond percolation.

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arXiv2025

Solidification estimates for random walks on supercritical percolation clusters

We consider the simple random walk on the infinite cluster of a general class of percolation models on $\mathbb{Z}^d$, $d\geq 3$, including Bernoulli percolation as well as models with strong, algebraically decaying correlations. For almost every realization of the percolation configuration, we obtain uniform controls on the absorption probability of a random walk by certain "porous interfaces" surrounding the discrete blow-up of a compact set $A$. These controls substantially generalize previous results obtained in arXiv:1706.07229 for Brownian motion in $\mathbb{R}^d$ and in arXiv:2012.05230 for random walks on $\mathbb{Z}^d$ equipped with uniformly elliptic edge weights to a manifestly non-elliptic framework.

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arXiv2024

Quantitative equilibrium fluctuations for interacting particle systems

We consider a class of interacting particle systems in continuous space of non-gradient type, which are reversible with respect to Poisson point processes with constant density. For these models, a rate of convergence was recently obtained in 10.1214/22-AOP1573 for certain finite-volume approximations of the bulk diffusion matrix. Here, we show how to leverage this to obtain quantitative versions of a number of results capturing the large-scale fluctuations of these systems, such as the convergence of two-point correlation functions and the Green-Kubo formula.

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arXiv2023

Lower bounds for bulk deviations for the simple random walk on $\mathbb{Z}^d$, $d\geq 3$

This article investigates the behavior of the continuous-time simple random walk on $\mathbb{Z}^d$, $d \geq 3$. We derive an asymptotic lower bound on the principal exponential rate of decay for the probability that the average value over a large box of some non-decreasing local function of the field of occupation times of the walk exceeds a given positive value. This bound matches at leading order the corresponding upper bound derived by Sznitman in arXiv:1906.05809, and is given in terms of a certain constrained minimum of the Dirichlet energy of functions on $\mathbb{R}^d$ decaying at infinity. Our proof utilizes a version of tilted random walks, a model originally constructed by Li in arXiv:1412.3959 to derive lower bounds on the probability of the event that the trace of a simple random walk disconnects a macroscopic set from an enclosing box.

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arXiv2023

Phase transition for level-set percolation of the membrane model in dimensions $d \geq 5$

We consider level-set percolation for the Gaussian membrane model on $\mathbb{Z}^d$, with $d \geq 5$, and establish that as $h \in \mathbb{R}$ varies, a non-trivial percolation phase transition for the level-set above level $h$ occurs at some finite critical level $h_\ast$, which we show to be positive in high dimensions. Along $h_\ast$, two further natural critical levels $h_{\ast\ast}$ and $\overline{h}$ are introduced, and we establish that $-\infty <\overline{h} \leq h_\ast \leq h_{\ast\ast} < \infty$, in all dimensions. For $h > h_{\ast\ast}$, we find that the connectivity function of the level-set above $h$ admits stretched exponential decay, whereas for $h < \overline{h}$, chemical distances in the (unique) infinite cluster of the level-set are shown to be comparable to the Euclidean distance, by verifying conditions identified by Drewitz, Ráth and Sapozhnikov, see arXiv:1212.2885, for general correlated percolation models. As a pivotal tool to study its level-set, we prove novel decoupling inequalities for the membrane model.

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arXiv2022

Absence of weak disorder for directed polymers on supercritical percolation clusters

We study the directed polymer model on infinite clusters of supercritical Bernoulli percolation containing the origin in dimensions $d \geq 3$, and prove that for almost every realization of the cluster and every strictly positive value of the inverse temperature, the polymer is in a strong disorder phase, answering a question from Cosco, Seroussi, and Zeitouni, see arXiv:2010.09503.

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arXiv2022

Smoothness of the diffusion coefficients for particle systems in continuous space

For a class of particle systems in continuous space with local interactions, we show that the asymptotic diffusion matrix is an infinitely differentiable function of the density of particles. Our method allows us to identify relatively explicit descriptions of the derivatives of the diffusion matrix in terms of correctors.

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arXiv2021

Disconnection and entropic repulsion for the harmonic crystal with random conductances

We study level-set percolation for the harmonic crystal on $\mathbb{Z}^d$, $d \geq 3$, 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 $α$ 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.

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arXiv2020

Solidification of porous interfaces and disconnection

In this article we obtain uniform estimates on the absorption of Brownian motion by porous interfaces surrounding a compact set. An important ingredient is the construction of certain resonance sets, which are hard to avoid for Brownian motion starting in the compact set. As an application of our results, we substantially strengthen the results of arXiv:1412.3960, and obtain when $d \ge 3$, large deviation upper bounds on the probability that simple random walk in $Z^d$, or random interlacements in $Z^d$, when their vacant set is in a strongly percolative regime, disconnect the discrete blow-up of a regular compact set from the boundary of the discrete blow-up of a box containing the compact set in its interior. Importantly, we make no convexity assumption on the compact set. It is plausible, although open at the moment, that the upper bounds that we derive in this work match in principal order the lower bounds of Xinyi Li and the second author (see arXiv:1310.2177) in the case of random interlacements, and of Xinyi Li (see arXiv:1412.3959) for the simple random walk.

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arXiv2019

Entropic repulsion for the Gaussian free field conditioned on disconnection by level-sets

We investigate level-set percolation of the discrete Gaussian free field on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the level-set of the Gaussian free field below a level $α$ disconnects the discrete blow-up of a compact set $A$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the Gaussian free field deviate from a specific multiple of the harmonic potential of $A$, when disconnection occurs. These bounds, combined with the findings of the recent article [12], show that conditionally on disconnection, the Gaussian free field experiences an entropic push-down proportional to the harmonic potential of $A$. In particular, due to the slow decay of correlations, the disconnection event affects the field on the whole lattice. Furthermore, we provide a certain 'profile' description for the field in the presence of disconnection. We show that while on a macroscopic scale the field is pinned around a level proportional to the harmonic potential of $A$, it locally retains the structure of a Gaussian free field shifted by a constant value. Our proofs rely crucially on the 'solidification estimates' developed in arXiv:1706.07229 by A.-S. Sznitman and the second author.

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arXiv2019

Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection

We investigate percolation of the vacant set of random interlacements on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the interlacement set at level $u$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^d$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the occupation times deviate from a specific function depending on the harmonic potential of $A$, when disconnection occurs. If certain critical levels coincide, which is plausible but open at the moment, these bounds imply that conditionally on disconnection, the occupation-time profile undergoes an entropic push governed by a specific function depending on $A$. Similar entropic repulsion phenomena conditioned on disconnection by level-sets of the discrete Gaussian free field on $\mathbb{Z}^d$, $d \geq 3$, have been obtained by the authors in arxiv:1808.09947. Our proofs rely crucially on the `solidification estimates' developed in arXiv:1706.07229 by A.-S. Sznitman and the second author.

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arXiv2018

Disconnection by level sets of the discrete Gaussian free field and entropic repulsion

We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $Z^d$, d bigger or equal to three, below a given level disconnects the discrete blow-up of a compact set A from the boundary of the discrete blow-up of a box that contains A, when the level set of the Gaussian free field above this level is in a strongly percolative regime. These bounds substantially strengthen the results of arXiv:1412.3960, where A was a box and the convexity of A played an important role in the proof. We also derive an asymptotic upper bound on the probability that the average of the Gaussian free field well inside the discrete blow-up of A is above a certain level when disconnection occurs. The derivation of the upper bounds uses the solidification estimates for porous interfaces that were derived in the work arXiv:1706.07229 of A.-S. Sznitman and the author to treat a similar disconnection problem for the vacant set of random interlacements. If certain critical levels for the Gaussian free field coincide, an open question at the moment, the asymptotic upper and lower bounds that we obtain for the disconnection probability match in principal order, and conditioning on disconnection lowers the average of the Gaussian free field well inside the discrete blow-up of A, which can be understood as entropic repulsion.

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