arXiv subjects
Publications and source records attributed to Dor Elboim.
We prove that the maximal dimension $d_N$ of an irreducible representation of the symmetric group $S_N$ satisfies $$d_N=\sqrt{N!} \, e^{-(\mathfrak{d}+o(1))\sqrt{N} }, \quad N\to \infty,$$ for some constant $\mathfrak{d}>0$. This answers a question raised by Vershik--Kerov in 1985.
We study the once-reinforced random walk on $\mathbb Z^d$, which is a self-interacting walk that has a higher probability to cross edges that were already visited. We prove that the walk is transient when $d\ge 6$ and when the reinforcement is small, establishing a conjecture of Sidoravicius in these dimensions. Moreover, in this case we prove that the walk behaves diffusively and can be coupled with Brownian motion. One of the main ideas in the proof is a certain capacity estimate which shows that the trajectory of the walk is nowhere heavy. We also use a game-theoretic-type ingredient that we call ``the demon" to force spatial independence in the process.
Consider a point process in Euclidean space obtained by perturbing the integer lattice with independent and identically distributed random vectors. Under mild assumptions on the law of the perturbations, we construct a translation-invariant perfect matching between this point process and the lattice, such that the matching distance has the same tail behavior as the hole probability of the point process, which is a natural lower bound.
We study first-passage percolation on $\mathbb Z ^2$ with independent and identically distributed weights, whose common distribution is uniform on $\{a,b\}$ with $0<a<b<\infty $. Following Ahlberg and De la Riva, we consider the passage time $\tau (n,k)$ of the minimal left-right crossing of the square $[0,n]^2$, whose vertical fluctuations are bounded by $k$. We prove that when $k\le n^{1/2-\epsilon}$, the event that $\tau (n,k)$ is larger than its median is noise sensitive. This improves the main result of Ahlberg and De la Riva which holds when $k\le n^{1/22-\epsilon }$. Under the additional assumption that the limit shape is not a polygon with a small number of sides, we extend the result to all $k\le n^{1-\epsilon }$. This extension follows unconditionally when $a$ and $b$ are sufficiently close. Under a stronger curvature assumption, we extend the result to all $k\le n$. This in particular captures the noise sensitivity of the event that the minimal left-right crossing $T_n=\tau (n,n)$ is larger than its median. Finally, under the curvature assumption, our methods give a lower bound of $n^{1/4-\epsilon }$ for the variance of the passage time $T_n$ of the minimal left-right crossing of the square. We prove the last bound also for absolutely continuous weight distributions, generalizing a result of Damron--Houdr\'e--\"Ozdemir, which holds only for the exponential distribution. Our approach differs from the previous works mentioned above; the key idea is to establish a small ball probability estimate in the tail by perturbing the weights for tail events using a Mermin--Wagner type estimate.
In the Lorentz mirror walk in dimension $d\geq 2$, mirrors are randomly placed on the vertices of $\mathbb{Z}^d$ at density $p\in[0,1]$. A light ray is then shot from the origin and deflected through the various mirrors in space. The object of study is the random trajectory obtained in this way, and it is of upmost interest to determine whether these trajectories are localized (finite) or delocalized (infinite). A folklore conjecture states that for $d=2$ these trajectories are finite for any density $p>0$, while in dimensions $d\geq 3$ and for $p>0$ small enough some trajectories are infinite. In this paper we prove that for all dimensions $d\geq 4$ and any small density $p$, the trajectories behave diffusively at all polynomial time scales $t\approx p^{-M}$ with $M>1$, and in particular, they do not close by this time.
In several settings (e.g., sensor networks and social networks), nodes of a graph are equipped with initial opinions, and the goal is to estimate the average of these opinions using local operations. A natural algorithm to achieve this is the edge-averaging process, where edges are repeatedly selected at random (according to independent Poisson clocks) and the opinions on the nodes of each selected edge are replaced by their average. The effectiveness of this algorithm is determined by its convergence rate. It is known that on a finite graph of $n$ nodes, the opinions reach approximate consensus in polynomial time. We prove that the convergence is much faster when the initial opinions are disordered (independent identically distributed): the time to reach approximate consensus is $O (\log^2n)$, and this bound is sharp. For infinite graphs, we show that for every $p\geq 1$, if the initial opinions are in $L^p$, then the opinion at each vertex converges to the mean in $L^p$, and if $p>4$, then almost sure convergence holds as well.
A minimal surface in a random environment (MSRE) is a $d$-dimensional surface in $(d+n)$-dimensional space which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. In this work, we consider the case of strongly correlated environments, realized by the model of harmonic MSRE in a fractional Brownian environment of Hurst parameter $H\in(0,1)$. This includes the case of Brownian environment ($H=1/2$ and $n=1$), which is commonly used to approximate the domain walls of the $(d+1)$-dimensional random-field Ising model. We prove that surfaces of dimension $d\in\{1,2,3\}$ delocalize with power-law fluctuations, and determine their precise transversal and minimal energy fluctuation exponents, as well as the stretched exponential exponents governing the tail decay of their distributions. These exponents are found to be the same in all codimensions $n$, depending only on $d$ and $H$. The transversal and minimal energy fluctuation exponents are specified by two scaling relations. We further show that surfaces of dimension $d=4$ delocalize with sub-power-law fluctuations, with their height and minimal energy fluctuations tied by a scaling relation. Lastly, we prove that surfaces of dimensions $d\ge 5$ localize. These results put several predictions from the physics literature on mathematically rigorous ground.
A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary conditions. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of $d$-dimensional minimal surfaces in a $(d+n)$-dimensional random environment. Specializing to a model that we term harmonic MSRE, in an ``independent'' random environment, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as versions of the scaling relation $χ=2ξ+d-2$ that ties together these two types of fluctuations. In particular, we prove, for all values of $n$, that the surfaces are delocalized in dimensions $d\le 4$ and localized in dimensions $d\ge 5$. Moreover, the surface delocalizes with power-law fluctuations when $d\le 3$ and sub-power-law fluctuations when $d=4$. Our localization results apply also to harmonic minimal surfaces in a periodic random environment.
We study planar first-passage percolation with independent weights whose common distribution is supported in $(0,\infty)$ and is absolutely continuous with respect to Lebesgue measure. We prove that the passage time from $x$ to $y$ denoted by $T(x,y)$ satisfies $$\max _{a\ge 0} \mathbb P \big( T(x,y)\in [a,a+1] \big) \le \frac{C}{\sqrt{\log \|x-y\|}},$$ answering a question posed by Ahlberg and de la Riva. This estimate recovers earlier results on the fluctuations of the passage time by Newman--Piza, Pemantle--Peres, and Chatterjee.
Random integers, sampled uniformly from $[1,x]$, share similarities with random permutations, sampled uniformly from $S_n$. These similarities include the Erdős--Kac theorem on the distribution of the number of prime factors of a random integer, and Billingsley's theorem on the largest prime factors of a random integer. In this paper we extend this analogy to non-uniform distributions. Given a multiplicative function $α\colon \mathbb{N} \to \mathbb{R}_{\ge 0}$, one may associate with it a measure on the integers in $[1,x]$, where $n$ is sampled with probability proportional to the value $α(n)$. Analogously, given a sequence $\{ θ_i\}_{i \ge 1}$ of non-negative reals, one may associate with it a measure on $S_n$ that assigns to a permutation a probability proportional to a product of weights over the cycles of the permutation. This measure is known as the generalized Ewens measure. We study the case where the mean value of $α$ over primes tends to some positive $θ$, as well as the weights $α(p) \approx (\log p)^γ$. In both cases, we obtain results in the integer setting which are in agreement with those in the permutation setting.
We analyze the asynchronous version of the DeGroot dynamics: In a connected graph $G$ with $n$ nodes, each node has an initial opinion in $[0,1]$ and an independent Poisson clock. When a clock at a node $v$ rings, the opinion at $v$ is replaced by the average opinion of its neighbors. It is well known that the opinions converge to a consensus. We show that the expected time $\mathbb E(τ_\varepsilon)$ to reach $\varepsilon$-consensus is poly$(n)$ in undirected graphs and in Eulerian digraphs, but for some digraphs of bounded degree it is exponential. Our main result is that in undirected graphs and Eulerian digraphs, if the degrees are uniformly bounded and the initial opinions are i.i.d., then $\mathbb E(τ_\varepsilon)=\text{polylog}(n)$ for every fixed $\varepsilon>0$. We give sharp estimates for the variance of the limiting consensus opinion, which measures the ability to aggregate information (``wisdom of the crowd''). We also prove generalizations to non-reversible Markov chains and infinite graphs. New results of independent interest on fragmentation processes and coupled random walks are crucial to our analysis.
In the interchange process on a graph $G=(V,E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $π_β:V\to V$ is formed for any time $β>0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. We prove the existence of infinite cycles in the interchange process on $\mathbb Z ^d$ for all dimensions $d\ge 5$ and all large $β$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions. In our proof, we study a self-interacting random walk called the cyclic time random walk. Using a multiscale induction we prove that it is diffusive and can be coupled with Brownian motion. One of the key ideas in the proof is establishing a local escape property which shows that the walk will quickly escape when it is entangled in its history in complicated ways.
We consider first-passage percolation on $\mathbb Z^2$ with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result leads to a quantitative resolution of the Benjamini--Kalai--Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge. We further prove that the limit shape assumption is satisfied for a specific family of distributions. Lastly, related to the 1965 Hammersley--Welsh highways and byways problem, we prove that the expected fraction of the square $\{-n,\dots ,n\}^2$ which is covered by infinite geodesics starting at the origin is at most an inverse power of $n$. This result is obtained without explicit limit shape assumptions.
Georgiou, Katkov and Tsodyks considered the following random process. Let $x_1,x_2,\ldots $ be an infinite sequence of independent, identically distributed, uniform random points in $[0,1]$. Starting with $S=\{0\}$, the elements $x_k$ join $S$ one by one, in order. When an entering element is larger than the current minimum element of $S$, this minimum leaves $S$. Let $S(1,n)$ denote the content of $S$ after the first $n$ elements $x_k$ join. Simulations suggest that the size $|S(1,n)|$ of $S$ at time $n$ is typically close to $n/e$. Here we first give a rigorous proof that this is indeed the case, and that in fact the symmetric difference of $S(1,n)$ and the set $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ is of size at most $\tilde{O}(\sqrt n)$ with high probability. Our main result is a more accurate description of the process implying, in particular, that as $n$ tends to infinity $ n^{-1/2}\big( |S(1,n)|-n/e \big) $ converges to a normal random variable with variance $3e^{-2}-e^{-1}$. We further show that the dynamics of the symmetric difference of $S(1,n)$ and the set $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ converges with proper scaling to a three dimensional Bessel process.
Let $Q^d$ be the $d$-dimensional binary hypercube. We say that $P=\{v_1,\ldots, v_k\}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\in [k-1]$ the edge $v_iv_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to a one coordinate in $v_{i+1}$. Form a random subgraph $Q^d_p$ by retaining each edge in $E(Q^d)$ independently with probability $p$. We show that there is a phase transition with respect to the length of a longest increasing path around $p=\frac{e}{d}$. Let $α$ be a constant and let $p=\fracα{d}$. When $α e$, whp there is a path of length $d-2$ in $Q^d_p$, and in fact, whether it is of length $d-2, d-1$, or $d$ depends on whether the all-zero and all-one vertices percolate or not.
Contrastive learning is a highly successful technique for learning representations of data from labeled tuples, specifying the distance relations within the tuple. We study the sample complexity of contrastive learning, i.e. the minimum number of labeled tuples sufficient for getting high generalization accuracy. We give tight bounds on the sample complexity in a variety of settings, focusing on arbitrary distance functions, both general $\ell_p$-distances, and tree metrics. Our main result is an (almost) optimal bound on the sample complexity of learning $\ell_p$-distances for integer $p$. For any $p \ge 1$ we show that $\tilde Θ(\min(nd,n^2))$ labeled tuples are necessary and sufficient for learning $d$-dimensional representations of $n$-point datasets. Our results hold for an arbitrary distribution of the input samples and are based on giving the corresponding bounds on the Vapnik-Chervonenkis/Natarajan dimension of the associated problems. We further show that the theoretical bounds on sample complexity obtained via VC/Natarajan dimension can have strong predictive power for experimental results, in contrast with the folklore belief about a substantial gap between the statistical learning theory and the practice of deep learning.
We study first-passage percolation on $\mathbb Z^d$, $d\ge 2$, with independent weights whose common distribution is compactly supported in $(0,\infty)$ with a uniformly-positive density. Given $ε>0$ and $v\in\mathbb Z^d$, which edges have probability at least $ε$ to lie on the geodesic between the origin and $v$? It is expected that all such edges lie at distance at most some $r(ε)$ from either the origin or $v$, but this remains open in dimensions $d\ge 3$. We establish the closely-related fact that the number of such edges is at most some $C(ε)$, uniformly in $v$. In addition, we prove a quantitative bound, allowing $ε$ to tend to zero as $\|v\|$ tends to infinity, showing that there are at most $O\big(ε^{-\frac{2d}{d-1}}(\log \|v\|)^C\big)$ such edges, uniformly in $ε$ and $v$. The latter result addresses a problem raised by Benjamin-Kalai-Schramm (2003). Our technique further yields a strengthened version of a lower bound on transversal fluctuations due to Licea-Newman-Piza (1996).
We show that for any set $A\subseteq [0,1]^n$ with $\text{Vol}(A)\ge 1/2$ there exists a line $\ell $ such that the one-dimensional Lebesgue measure of $\ell \cap A$ is at least $Ω( n^{1/4} )$. The exponent $1/4$ is tight. More generally, for a probability measure $μ$ on $\mathbb R ^n$ and $0<a<1$ define \begin{equation*} L(μ,a):= \inf_{A ; μ(A) = a} \sup _{\ell \text{ line}} \big| \ell \cap A\big| \end{equation*} where $|\cdot | $ stands for the one-dimensional Lebesgue measure. We study the asymptotic behavior of $L(μ,a)$ when $μ$ is a product measure and when $μ$ is the uniform measure on the $\ell _p$ ball. We observe a rather unified behavior in a large class of product measures. On the other hand, for $\ell_p$ balls with $1 \leq p \leq \infty$ we find that there are phase transitions of different types.