These are expanded lecture notes for a minicourse taught at the "School on disordered media" at the Alfred Renyi institute in Budapest, January 2025.

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 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.

We study asymptotic laws of random walks on $\mathbb Z^d$ ($d\ge1$) in deterministic reversible environments defined by an assignment of a positive and finite conductance to each edge of $\mathbb Z^d$. The walk jumps across an edge with probability proportional to its conductance. We identify a deterministic set of conductance configurations for which an Invariance Principle (i.e., convergence in law to Brownian motion under diffusive scaling of space and time) provably holds. This set is closed under translations and zero-density perturbations and carries all ergodic conductance laws subject to certain moment conditions. The proofs are based on martingale approximations whose control relies on 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 proofs of "deterministic homogenization" in other motions in disordered media.

Nous considérons le champ libre gaussien discret (DGFF) dans des domaines DN⊆Z2 qu’on obtient, via une mise à l’échelle par N, à partir de domaines raisonnables D⊆R2. Nous étudions les statistiques des valeurs d’ordre logN en dessous du maximum absolu. Encodés dans un processus ponctuel sur D×R, la distribution spatiale mise à l’échelle de ces ensembles de niveaux proches du maximum sur D N et les valeurs du champ (en unités de logN en dessous du maximum absolu) convergent, pour N→∞, en loi vers le produit de la gravité quantique critique de Liouville (cLQG) ZD et de la loi de Rayleigh. La convergence est valable conjointement avec le processus extrémal, pour lequel ZD représente la mesure d’intensité du processus ponctuel de Poisson qui apparait à la limite, et avec le DGFF lui-même ; le cLQG défini par le champ limite coïncide alors avec ZD. Bien que le processus limite des valeurs proches du maximum soit mesurable par rapport au GFF (continu) limite, le processus limite des valeurs proches du maximum ne l’est pas. Nos résultats expliquent pourquoi les différentes manières de régulariser la mesure cLQG du réseau conduisent au même objet limite, à une normalisation globale près.

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.

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 $\beta|x-y|^{-s}$, for $s\in(d,2d)$, $\beta>0$ and $|\cdot|$ a norm on $\mathbb R^d$. We first show that, for all but a countably many $\beta>0$, 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 $\phi_\beta(r)(\log r)^\Delta$, where $\Delta^{-1}:=\log_2(2d/s)$ and $\phi_\beta$ is a positive, bounded and continuous function subject to $\phi_\beta(r^\gamma)=\phi_\beta(r)$ for $\gamma:=s/(2d)$. 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 $\phi_\beta$ is constant which we show to be false by proving that $(\log\beta)^\Delta\phi_\beta$ tends, as $\beta\to\infty$, 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.

We consider the Discrete Gaussian Free Field (DGFF) in domains DN subset of Z2 arising, via scaling by N, from nice domains D subset of R2. We study the statistics of the values order ./log N below the absolute maximum. Encoded as a point process on D x R, the scaled spatial distribution of these near-extremal level sets in DN and the field values (in units of ./log N below the absolute maximum) tends, as N -> infinity, in law to the product of the critical Liouville Quantum Gravity (cLQG) ZD and the Rayleigh law. The convergence holds jointly with the extremal process, for which ZD 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 ZD. 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.

Given a square box $\Lambda_n\subseteq\mathbb Z^2$ of side-length $L^n$ with $L,n>1$, we study hierarchical random fields $\{\phi_x\colon x\in\Lambda_n\}$ with law proportional to ${\rm e}^{\frac12\beta(\varphi,\Delta_n\phi)}\prod_{x\in\Lambda_n}\nu({\rm d}\phi_x)$, where $\beta>0$ is the inverse temperature, $\Delta_n$ is a hierarchical Laplacian on $\Lambda_n$, and $\nu$ 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 $\langle\phi_x\phi_y\rangle$ and the fractional charge $\langle{\rm e}^{2\pi{\rm i}\alpha(\phi_x-\phi_y)}\rangle$ in the subcritical $\beta<\beta_{\rm c}:=\pi^2/\log L$, critical $\beta=\beta_{\rm c}$ and slightly supercritical $\beta>\beta_{\rm c}$ regimes. The field exhibits logarithmic correlations throughout albeit with a distinct $\beta$-dependence of the variance and fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point.

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.

Given a square box Λ_n⊆ℤ^2 of side-length L^n with L,n>1, we study hierarchical random fields {ϕ_x x∈Λ_n} with law proportional to e^1/2β(φ,Δ_nϕ)∏_x∈Λ_nν( dϕ_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 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→∞, for the covariance ⟨ϕ_xϕ_y⟩ and the fractional charge ⟨ e^2π iα(ϕ_x-ϕ_y)⟩ in the subcritical β<β_ c:=π^2/log L, critical β=β_ c and slightly supercritical β>β_ c regimes. The field exhibits logarithmic correlations throughout albeit with a distinct β-dependence of the variance and fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point.

Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing-point depression upon freezing of solutions. Specifically, we devise an Ising-based model of a solvent--solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. The boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezing-point depression. The limit of infinitesimal concentrations is described in a subsequent paper.

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.

We study the extremal properties of the "integer-valued Gaussian" a.k.a.\ DG-model on the hierarchical lattice $\Lambda_n:=\{1,\dots,b\}^n$ (with $b\ge2$) of depth $n$. This is a random field $\varphi\in\mathbb Z^{\Lambda_n}$ with law proportional to $e^{\frac12\beta(\varphi,\Delta_n\varphi)}\prod_{x\in\Lambda_n}\#(d\varphi_x)$, where $\Delta_n$ is the hierarchical Laplacian, $\beta$ is the inverse temperature and $\#$ is the counting measure on $\mathbb Z$. Denoting $\beta_c:=2\pi^2/\log b$ and $m_n:=\beta^{-1/2}[(2\log b)^{1/2}n-\frac32(2\log b)^{-1/2}\log n]$, for $0<\beta<\beta_c$ 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 $\max_{x\in\Lambda_n}\varphi_x-\lfloor m_n\rfloor$ 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 $(0,\beta_c]$ marks the full range of values of $\beta$ for which the renormalization-group iterations tend to a "trivial" fixed point.

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.

The general topic of the mini-workshop New Horizons in Motions in Random Media was the study of random walks in random environments, both in their own right and in relation to stochastic homogenization and to models in statistical mechanics, in particular spin system. This is a subject at the intersection of probability, analysis and mathematical physics, and the workshop brought together leading researchers from those areas. While each of these areas has been quite active for decades with many remarkable breakthroughs obtained throughout the years, the workshop provided a unique opportunity to identify principal new objectives and initiate new collaborations.

We continue our study of colligative properties of solutions initiated in ref. 1. We focus on the situations where, in a system of linear size L, the concentration and the chemical potential scale like c=ξ/L and h=b/L, respectively. We find that there exists a critical value ξt such that no phase separation occurs for ξ≤ξt while, for ξ>ξt, the two phases of the solvent coexist for an interval of values of b. Moreover, phase separation begins abruptly in the sense that a macroscopic fraction of the system suddenly freezes (or melts) forming a crystal (or droplet) of the complementary phase when b reaches a critical value. For certain values of system parameters, under “frozen” boundary conditions, phase separation also ends abruptly in the sense that the equilibrium droplet grows continuously with increasing b and then suddenly jumps in size to subsume the entire system. Our findings indicate that the onset of freezing-point depression is in fact a surface phenomenon.

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.

We study exceptional sets of the local time of the continuous-time simple random walk in scaled-up (by N ) versions D_N⊆ℤ^2 of bounded open domains D⊆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→∞ , 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_√(θ) . 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.

We study random walks on $\mathbb Z^d$ among random conductances $\{C_{xy}\colon x,y\in\mathbb Z^d\}$ that permit jumps of arbitrary length. Apart from joint ergodicity with respect to spatial shifts, we assume only that the nearest-neighbor conductances are uniformly positive and that $\sum_{x\in\mathbb Z^d} C_{0x}|x|^2$ is integrable. Our focus is on the Quenched Invariance Principle (QIP) which we establish in all $d\ge3$ by a combination of corrector methods and heat-kernel technology. In particular, a QIP thus holds for random walks on long-range percolation graphs with exponents larger than $d+2$ in all $d\ge3$, provided all nearest-neighbor edges are present. We then show that, for long-range percolation with exponents between $d+2$ and $2d$, the corrector fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in $d\ge4$ under the conditions close to, albeit not exactly, complementary to those of the recent work of S. Andres, M. Slowik and J.-D. Deuschel.

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.