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Marek Biskup

Publications and source records attributed to Marek Biskup.

At least 19 recordsLinked to original sources
arXiv2025

Homogenization theory of random walks among deterministic conductances

We study the asymptotic distribution of random walks on $\mathbb Z^d$ ($d\ge1$) in deterministic reversible environments defined by an assignment of a positive conductance to each edge of $\mathbb Z^d$. We identify a deterministic set of conductance configurations for which the walk obeys an Invariance Principle; i.e., converges in law to a non-degenerate Brownian motion under diffusive scaling of space and time. This set is closed under translations and zero-density perturbations and carries all ergodic conductance laws subject to certain moment conditions. The proofs rely on martingale approximations whose main step is the conversion of averages in time and physical space under the deterministic environment to those in a suitable stochastic counterpart. Our study sets up a framework for "de-randomized homogenization" of other motions in disordered media.

math.PR↗
arXiv2025

Extremal process of the local time of simple random walk on a regular tree

We study a continuous-time simple random walk on a regular rooted tree of depth $n$ in two settings: either the walk is started from a leaf vertex and run until the tree root is first hit or it is started from the root and run until it has spent a prescribed amount of time there. In both cases we show that the extremal process associated with centered square-root local time on the leaves tends, as $n\to\infty$, to a decorated Poisson point process with a random intensity measure. While the intensity measure is specific to the local-time problem at hand, the decorations are exactly those for the tree-indexed Markov chain (a.k.a. Branching Random Walk or Gaussian Free Field) with normal step distribution. The proof demonstrates the latter by way of a Lindeberg-type swap of the decorations of the two processes which itself relies on a well-known isomorphism theorem.

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arXiv2025

Phase transition and critical behavior in hierarchical integer-valued Gaussian and Coulomb gas models

Given a square box $Λ_n\subseteq\mathbb Z^2$ of side length $L^n$ with $L,n>1$, we study hierarchical random fields $\{ϕ_x\colon x\inΛ_n\}$ with law proportional to ${\rm e}^{\frac12β(ϕ,Δ_nϕ)}\prod_{x\inΛ_n}ν({\rm d}ϕ_x)$, where $β>0$ is the inverse temperature, $Δ_n$ is a hierarchical Laplacian on $Λ_n$, and $ν$ is a non-degenerate $1$-periodic measure on $\mathbb R$. Our setting includes the integer-valued Gaussian field (a.k.a. DG model or Villain Coulomb gas) and the sine-Gordon model. Relying on renormalization group analysis we derive sharp asymptotic formulas, in the limit as $n\to\infty$, for the covariance $\langleϕ_xϕ_y\rangle$ and the fractional charge $\langle {\rm e}^{2π{\rm i}α(ϕ_x-ϕ_y)}\rangle$ in the subcritical $β<β_{\rm c}:=π^2/\log L$, critical $β=β_{\rm c}$ and slightly supercritical $β>β_{\rm c}$ regimes. The field exhibits logarithmic correlations throughout albeit with a distinct $β$-dependence of both the covariance scale and the fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point.

math.PR↗
arXiv2024

On support sets of the critical Liouville Quantum Gravity

We study fractal properties of support sets of the critical Liouville Quantum Gravity (cLQG) associated with the Gaussian Free Field in planar domains. Specifically, we completely characterize the gauge functions $ϕ$ (subject to mild monotonicity conditions) for which the cLQG admits a support set of finite $ϕ$-Hausdorff measure. As a corollary, we settle the conjecture that the cLQG is supported on a set of vanishing Hausdorff dimension. Our proofs are based on the fact that the cLQG describes the near-critical level sets of Discrete Gaussian Free Field.

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arXiv2023

A limit law for the maximum of subcritical DG-model on a hierarchical lattice

We study the extremal properties of the "integer-valued Gaussian" a.k.a.\ DG-model on the hierarchical lattice $Λ_n:=\{1,\dots,b\}^n$ (with $b\ge2$) of depth $n$. This is a random field $φ\in\mathbb Z^{Λ_n}$ with law proportional to $e^{\frac12β(φ,Δ_nφ)}\prod_{x\inΛ_n}\#(dφ_x)$, where $Δ_n$ is the hierarchical Laplacian, $β$ is the inverse temperature and $\#$ is the counting measure on $\mathbb Z$. Denoting $β_c:=2π^2/\log b$ and $m_n:=β^{-1/2}[(2\log b)^{1/2}n-\frac32(2\log b)^{-1/2}\log n]$, for $0<β<β_c$ we prove that, along increasing sequences of $n$ such that the fractional part of $m_{n}$ converges to an $s\in[0,1)$, the centered maximum $\max_{x\inΛ_n}φ_x-\lfloor m_n\rfloor$ tends (as $n\to\infty$) in law to a discrete variant of a randomly shifted Gumbel law with the shift depending non-trivially on $s$. The convergence extends to the extremal process whose law tends to a decorated Poisson point process with a random intensity measure. The proofs rely on renormalization-group analysis which enables a tight coupling of the DG-model to a Gaussian Free Field. The interval $(0,β_c]$ marks the full range of values of $β$ for which the renormalization-group iterations tend to a "trivial" fixed point.

math.PR↗
arXiv2023

Near-maxima of the two-dimensional Discrete Gaussian Free Field

We consider the Discrete Gaussian Free Field (DGFF) in domains $D_N\subseteq\mathbb Z^2$ arising, via scaling by $N$, from nice domains $D\subseteq\mathbb R^2$. We study the statistics of the values order $\sqrt{\log N}$ below the absolute maximum. Encoded as a point process on $D\times\mathbb R$, the scaled spatial distribution of these near-extremal level sets in $D_N$ and the field values (in units of $\sqrt{\log N}$ below the absolute maximum) tends, as $N\to\infty$, in law to the product of the critical Liouville Quantum Gravity (cLQG) $Z^D$ and the Rayleigh law. The convergence holds jointly with the extremal process, for which $Z^D$ enters as the intensity measure of the limiting Poisson point process, and that of the DGFF itself; the cLQG defined by the limit field then coincides with $Z^D$. While the limit near-extremal process is measurable with respect to the limit continuum GFF, the limit extremal process is not. Our results explain why the various ways to "norm" the lattice cLQG measure lead to the same limit object, modulo overall normalization.

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arXiv2022

An invariance principle for one-dimensional random walks in degenerate dynamical random environments

We study random walks on the integers driven by a sample of time-dependent nearest-neighbor conductances that are bounded but are permitted to vanish over time intervals of positive Lebesgue-length. Assuming only ergodicity of the conductance law under space-time shifts and a moment assumption on the time to accumulate a unit conductance over a given edge, we prove that the walk scales, under a diffusive scaling of space and time, to a non-degenerate Brownian motion for a.e. realization of the environment. The conclusion particularly applies to random walks on one-dimensional dynamical percolation subject to fairly general stationary edge-flip dynamics.

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arXiv2022

Arithmetic oscillations of the chemical distance in long-range percolation on $\mathbb Z^d$

We consider a long-range percolation graph on $\mathbb Z^d$ where, in addition to the nearest-neighbor edges of $\mathbb Z^d$, distinct $x,y\in\mathbb Z^d$ are connected by an edge independently with probability asymptotic to $β|x-y|^{-s}$, for $s\in(d,2d)$, $β>0$ and $|\cdot|$ a norm on $\mathbb R^d$. We first show that, for all but a countably many $β>0$, the graph-theoretical (a.k.a. chemical) distance between typical vertices at $|\cdot|$-distance $r$ is, with high probability as $r\to\infty$, asymptotic to $ϕ_β(r)(\log r)^Δ$, where $Δ^{-1}:=\log_2(2d/s)$ and $ϕ_β$ is a positive, bounded and continuous function subject to $ϕ_β(r^γ)=ϕ_β(r)$ for $γ:=s/(2d)$. The proof parallels that in a continuum version of the model where a similar scaling was shown earlier by the first author and J. Lin. This work also conjectured that $ϕ_β$ is constant which we show to be false by proving that $(\logβ)^Δϕ_β$ tends, as $β\to\infty$, to a non-constant limit which is independent of the specifics of the model. The proof reveals arithmetic rigidity of the shortest paths that maintain a hierarchical (dyadic) structure all the way to unit scales.

math.PR↗
arXiv2022

Exceptional points of two-dimensional random walks at multiples of the cover time

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by $N$) versions $D_N\subseteq \mathbb Z^2$ of bounded open domains $D\subseteq \mathbb R^2$. Upon exit from $D_N$, the walk lands on a "boundary vertex" and then reenters $D_N$ through a random boundary edge in the next step. In the parametrization by the local time at the "boundary vertex" we prove that, at times corresponding to a $θ$-multiple of the cover time of $D_N$, the sets of suitably defined $λ$-thick (i.e., heavily visited) and $λ$-thin (i.e., lightly visited) points are, as $N\to\infty$, distributed according to the Liouville Quantum Gravity $Z^D_λ$ with parameter $λ$-times the critical value. For $θ<1$, also the set of avoided vertices (a.k.a. late points) and the set where the local time is of order unity are distributed according to $Z^D_{\sqrtθ}$. The local structure of the exceptional sets is described as well, and is that of a pinned Discrete Gaussian Free Field for the thick and thin points and that of random-interlacement occupation-time field for the avoided points. The results demonstrate universality of the Gaussian Free Field for these extremal problems.

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arXiv2021

A limit law for the most favorite point of simple random walk on a regular tree

We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. For the walk started from a leaf vertex and stopped upon hitting the root we prove that, in the limit as as the depth of the tree tends to infinity, the suitably scaled and centered maximal time spent at any leaf converges to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with square-root local-time process on the tree.

math.PR↗
arXiv2021

Quenched Invariance Principle for a class of random conductance models with long-range jumps

We study random walks on $\mathbb Z^d$ (with $d\ge 2$) among stationary ergodic random conductances $\{C_{x,y}\colon x,y\in\mathbb Z^d\}$ that permit jumps of arbitrary length. Our focus is on the Quenched Invariance Principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the $p$-th moment of $\sum_{x\in\mathbb Z^d}C_{0,x}|x|^2$ and $q$-th moment of $1/C_{0,x}$ for $x$ neighboring the origin are finite for some $p,q\ge1$ with $p^{-1}+q^{-1}<2/d$. In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than $2d$ in all $d\ge2$, provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between $d+2$ and $2d$, the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in $d\ge3$ under the conditions complementary to those of the recent work of P. Bella and M. Schäffner. These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.

math.PR↗
arXiv2020

Conformal symmetries in the extremal process of two-dimensional discrete Gaussian Free Field

We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions ($x$) and centered values ($h$) of the extreme local maxima in lattice versions of a bounded domain $D\subset\mathbb C$ converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure $Z^D(dx)\otimes e^{-αh}d h$, where $α$ is a constant and $Z^D$ is a random a.s.-finite measure on $D$. The random measures $\{Z^D\}$ are naturally interrelated; restrictions to subdomains are governed by a Gibbs-Markov property and images under analytic bijections $f$ by the transformation rule $(Z^{f(D)}\circ f)(d x)\overset{\text{law}}=|f'(x)|^4\, Z^D(d x)$. Conditions are given that determine the laws of these measures uniquely. These identify $Z^D$ with the critical Liouville Quantum Gravity associated with the Continuum Gaussian Free Field.

math.PR↗
arXiv2019

Exceptional points of discrete-time random walks in planar domains

Given a sequence of lattice approximations $D_N\subset\mathbb Z^2$ of a bounded continuum domain $D\subset\mathbb R^2$ with the vertices outside $D_N$ fused together into one boundary vertex $\varrho$, we consider discrete-time simple random walks in $D_N\cup\{\varrho\}$ run for a time proportional to the expected cover time and describe the scaling limit of the exceptional level sets of the thick, thin, light and avoided points. We show that these are distributed, up a spatially-dependent log-normal factor, as the zero-average Liouville Quantum Gravity measures in $D$. The limit law of the local time configuration at, and nearby, the exceptional points is determined as well. The results extend earlier work by the first two authors who analyzed the continuous-time problem in the parametrization by the local time at $\varrho$. A novel uniqueness result concerning divisible random measures and, in particular, Gaussian Multiplicative Chaos, is derived as part of the proofs.

math.PR↗
arXiv2019

Return probability and recurrence for the random walk driven by two-dimensional Gaussian free field

Given any $γ>0$ and for $η=\{η_v\}_{v\in \mathbb Z^2}$ denoting a sample of the two-dimensional discrete Gaussian free field on $\mathbb Z^2$ pinned at the origin, we consider the random walk on~$\mathbb Z^2$ among random conductances where the conductance of edge $(u, v)$ is given by $\mathrm{e}^{γ(η_u + η_v)}$. We show that, for almost every~$η$, this random walk is recurrent and that, with probability tending to~1 as $T\to \infty$, the return probability at time~$2T$ decays as $T^{-1+o(1)}$. In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius~$N$ scales as $N^{ψ(γ)+o(1)}$ with $ψ(γ)>2$ for all~$γ>0$. Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance~$N$ behaves as~$N^{o(1)}$.

math.PR↗
arXiv2019

Extrema of the two-dimensional Discrete Gaussian Free Field

These lecture notes offer a gentle introduction to the two-dimensional Discrete Gaussian Free Field with particular attention paid to the scaling limits of the level sets at heights proportional to the absolute maximum. The bulk of the text is based on recent joint papers with O. Louidor and with J. Ding and S. Goswami. Still, new proofs of the tightness and distributional convergence of the centered DGFF maximum are presented that by-pass the use of the modified Branching Random Walk. The text contains a wealth of instructive exercises and a list of open questions and conjectures for future research.

math.PR↗
arXiv2018

An invariance principle for one-dimensional random walks among dynamical random conductances

We study variable-speed random walks on $\mathbb Z$ driven by a family of nearest-neighbor time-dependent random conductances $\{a_t(x,x+1)\colon x\in\mathbb Z, t\ge0\}$ whose law is assumed invariant and ergodic under space-time shifts. We prove a quenched invariance principle for the random walk under the minimal moment conditions on the environment; namely, assuming only that the conductances possess the first positive and negative moments. A novel ingredient is the representation of the parabolic coordinates and the corrector via a dual random walk which is considerably easier to analyze.

math.PR↗
arXiv2018

Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field

We study the local structure of the extremal process associated with the Discrete Gaussian Free Field (DGFF) in scaled-up (square-)lattice versions of bounded open planar domains subject to mild regularity conditions on the boundary. We prove that, in the scaling limit, this process tends to a Cox process decorated by independent, correlated clusters whose distribution is completely characterized. As an application, we control the scaling limit of the discrete supercritical Liouville measure, extract a Poisson-Dirichlet statistics for the limit of the Gibbs measure associated with the DGFF and establish the "freezing phenomenon" conjectured to occur in the "glassy" phase. In addition, we prove a local limit theorem for the position and value of the absolute maximum. The proofs are based on a concentric, finite-range decomposition of the DGFF and entropic-repulsion arguments for an associated random walk. Although we naturally build on our earlier work on this problem, the methods developed here are largely independent.

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