arXiv subjects

Oren Louidor

Publications and source records attributed to Oren Louidor.

At least 19 recordsLinked to original sources

arXiv2026

Self-Stabilizing Algorithms in the Uniform Port Model

We introduce a distributed computational model referred to as the \emph{uniform port} model. An algorithm operating in this model is defined by means of local automata associated with the ports (a.k.a.\ half-edges) of the input graph. The crux of the uniform port model is that a single constant-size finite automaton is hosted by every port of every graph, making the model \emph{truly uniform}. Moreover, since the new model explicitly supports the assignment of (input and) output labels to the graph's (half-)edges, it facilitates natural formulations of (half-)edge-labeling problems such as maximal matching and sinkless orientation, which are outside the expressivity scope of prior node-centric truly uniform distributed computational models. The main technical contribution of this paper is the design of efficient (i.e., with poly-logarithmic runtime) \emph{self-stabilizing} uniform port algorithms, operating on general graphs, for various fundamental local symmetry breaking problems, including maximal independent set, maximal matching, sinkless orientation, and maximal node/edge $k$-coloring. While efficient self-stabilizing algorithms for local symmetry breaking problems have been extensively studied in stronger computational models, our work is the first to demonstrate the existence of such algorithms in a truly uniform model.

cs.DC↗

arXiv2026

Aging in a spin glass with logarithmic correlations

We consider a continuous-time random walk on the discrete two dimensional box, driven by the discrete Gaussian Free Field (DGFF) acting as potential: When at a vertex, the walk waits an exponentially distributed time with mean given by the exponential of the field times an inverse temperature parameter and then jumps to one of its neighbors uniformly at random. We prove that when the temperature is below the critical value the walk exhibits ``aging'' at a range of pre-equilibrium time scales: Observed at any such time and then again after an additional time of the same order, there is a positive probability that the walk is found within finite distance from where it was before, with this probability given asymptotically by the Generalized Arcsine Law with a temperature-dependent parameter. We show that this is a consequence of an intricate trapping mechanism which localizes the walk for periods of time which increase with the age of the system, and describe the complex structure of the underlying trapping landscape, which is intimately related to the geometry of the near-extreme level-sets of the DGFF. Altogether, this work demonstrates for the first time an Arcsine-Law aging in the case of a spin-glass-type system with a logarithmically correlated potential, throughout its glassy phase, as predicted in the physic literature.

math.PR↗

arXiv2026

A Limit in Law for the Cover Time and Last Visited Vertex of Wired Planar Domains

We derive a scaling limit in law for the cover time of a simple random walk on a lattice version of a scaled-up planar domain with wired boundary conditions. The limiting distribution is that of a Gumbel Random Variable shifted randomly by an independent quantity which is equal to the full mass of a variant of the critical Liouville Quantum Gravity Measure on the same domain. We also derive a limit in law for the scaled location of the vertex visited last by the walk. Here the limit turns out to be precisely the critical Liouville Measure, normalized by its total mass. Both limits hold jointly with the limiting joint law explicitly described. These results resolve well known open problems in the field, in the case of wired boundary conditions. The proof is based on comparison with the extremal landscape of the discrete Gaussian Free Field, and in particular a version there-of obtained by conditioning the average value of the field to be zero.

math.PR↗

arXiv2026

Second Order Asymptotics for the Hard Wall Probability of the 2D Harmonic Crystal

We estimate the probability that the discrete Gaussian free field on a planar domain with Dirichlet boundary conditions stays positive in the bulk. Improving upon the result by Bolthausen, Deuschel and Giacomin from 2001, we derive the order of the subleading term of this probability when a sequence of discretized scale-ups of given domain and compactly included smooth bulk are considered. A main ingredient in the proof is the double exponential decay of the right tail of the centered minimum of the field in the bulk, conditioned on a certain weighted average of its values to be zero.

math.PR↗

arXiv2025

Learning words in groups: fusion algebras, tensor ranks and grokking

In this work, we demonstrate that a simple two-layer neural network with standard activation functions can learn an arbitrary word operation in any finite group, provided sufficient width is available and exhibits grokking while doing so. To explain the mechanism by which this is achieved, we reframe the problem as that of learning a particular $3$-tensor, which we show is typically of low rank. A key insight is that low-rank implementations of this tensor can be obtained by decomposing it along triplets of basic self-conjugate representations of the group and leveraging the fusion structure to rule out many components. Focusing on a phenomenologically similar but more tractable surrogate model, we show that the network is able to find such low-rank implementations (or approximations thereof), thereby using limited width to approximate the word-tensor in a generalizable way. In the case of the simple multiplication word, we further elucidate the form of these low-rank implementations, showing that the network effectively implements efficient matrix multiplication in the sense of Strassen. Our work also sheds light on the mechanism by which a network reaches such a solution under gradient descent.

cs.LG↗

arXiv2025

Wetting Transition on Trees I: Percolation With Clustering

A new ``Percolation with Clustering'' (PWC) model is introduced, where (the probabilities of) site percolation configurations on the leaf set of a binary tree are rewarded exponentially according to a generic function, which measures the degree of clustering in the configuration. Conditions on such ``clustering function'' are given for the existence of a limiting free energy and a wetting transition, namely the existence of a non-trivial percolation parameter threshold above and only above which the set of ``dry'' (open) sites have an asymptotic density. Several examples of clustering functions are given and studied using the general theory. The results here will be used in a sequel paper to study the wetting transition for the discrete Gaussian free field on the tree subject to a hard wall constraint.

math.PR↗

arXiv2025

A Probabilistic Proof for Stable Fluctuations in the Extremal Process of Branching Brownian Motion

We give a probabilistic proof for the emergence of the Stable-$1$ Law for the random fluctuations of the mass of the extremal process of branching Brownian Motion away from its tip. This result was already shown by Mytnik et al. albeit using PDE techniques. As a consequence, we demystify the origin of these fluctuations and the meaning of the deterministic centering function required.

math.PR↗

arXiv2024

Gaussian free field on the tree subject to a hard wall II: Asymptotics

This is the second in a series of two works which study the discrete Gaussian free field on the binary tree when all leaves are conditioned to be positive. In the first work ("Gaussian free field on the tree subject to a hard wall I: Bounds") we identified the repulsion profile followed by the field in order to fulfill this "hard-wall constraint" event. In this work, we use these findings to obtain a comprehensive, sharp asymptotic description of the law of the field under this conditioning. We provide asymptotics for both local statistics, namely the (conditional) law of the field in a neighborhood of a vertex, as well as global statistics, including the (conditional) law of the minimum, maximum, empirical population mean and all subcritical exponential martingales. We conclude that the laws of the conditional and unconditional fields are asymptotically mutually singular with respect to each other.

math.PR↗

arXiv2024

Gaussian free field on the tree subject to a hard wall I: Bounds

This is the first in a series of two works which study the discrete Gaussian free field on the binary tree when all leaves are conditioned to be positive. In this work, we obtain sharp asymptotics for the probability of this "hard-wall constraint" event, and identify the repulsion profile followed by the field in order to achieve it. We also provide estimates for the mean, fluctuations and covariances of the field under the conditioning, which show that in the first log-many generations the field is localized around its mean. These results are used in the sequel work ("Gaussian free field on the tree subject to a hard wall II: Asymptotics") to obtain a comprehensive asymptotic description of the law of the field under the conditioning.

math.PR↗

arXiv2024

Tightness for the Cover Time of Wired Planar Domains

We consider a continuous time simple random walk on a subset of the square lattice with wired boundary conditions: the walk transitions at unit edge rate on the graph obtained from the lattice closure of the subset by contracting the boundary into one vertex. We study the cover time of such walk, namely the time it takes for the walk to visit all vertices in the graph. Taking a sequence of subsets obtained as scaled lattice versions of a nice planar domain, we show that the square root of the cover time normalized by the size of the subset, is tight around $\frac{1}{\sqrtπ} \log N - \frac{1}{4 \sqrtπ} \log \log N$, where $N$ is the scale parameter. This proves an analog, for the wired case, of a conjecture by Bramson and Zeitouni from 2009. The proof is based on comparison with the extremal landscape of the discrete Gaussian free field.

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.

math.PR↗

arXiv2024

On the Growth of the Extremal and Cluster Level Sets in Branching Brownian Motion

We study the limiting extremal and cluster point processes of branching Brownian motion. The former records the heights of all extreme values of the process, while the latter records the relative heights of extreme values in a genealogical neighborhood of order unity around a local maximum thereof. For the extremal point process, we show that the mass of upper level sets $[-v, \infty)$ grows as $C_\star Z v e^{\sqrt{2} v}(1+o(1))$ as $v \to \infty$, almost surely, where $Z$ is the limit of the associated derivative martingale and $C_\star \in (0, \infty)$ is a universal constant. For the cluster point process, we show that the logarithm of the mass of $[-v, \infty)$ grow as $\sqrt{2}v$ minus random fluctuations of order $v^{2/3}$, which are governed by an explicit law in the limit. The first result improves upon the works of Cortines et al. (arXiv:1703.06529) and Mytnik et al. (arXiv:2009.02042) in which asymptotics are shown in probability, while the second makes rigorous the derivation in the physics literature by Mueller et al. (arXiv:1910.06382) and Le et al. (arXiv:2207.07672) and resolves a conjecture thereof.

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.

math.PR↗

arXiv2022

A toy model for DLA arm growth in a wedge

In this paper, we consider a non-homogeneous discrete-time Markov chain which can be seen as a toy model for the growth of the arms of the DLA (Diffusion limited aggregation) process in a sub-linear wedge. It is conjectured that in a thin enough linear wedge there is only one infinite arm in the DLA cluster and we demonstrate this phenomenon in our model. The technique follows a bootstrapping argument, in which we iteratively prove ever faster growth rate.

math.PR↗

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↗

arXiv2020

Ballot Theorems for the Two-Dimensional Discrete Gaussian Free Field

We provide uniform bounds and asymptotics for the probability that a two-dimensional discrete Gaussian free field on an annulus-like domain and with Dirichlet boundary conditions stays negative as the ratio of the radii of the inner and the outer boundary tends to infinity.

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

The Structure of Extreme Level Sets in Branching Brownian Motion

We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida, 2011. The proofs rely on studying the cluster distribution and should carry over to the branching random walk and the two-dimensional discrete Gaussian free field with no conceptual difficulty.

math.PR↗