arXiv subjects

Dylan J. Altschuler

Publications and source records attributed to Dylan J. Altschuler.

18 recordsLinked to original sources

arXiv2026

Long-range expanders: construction and cutoff

Long-range expansion is a tree-like volume growth condition on bounded-degree regular graphs, introduced by Dodos, Tikhomirov, Tyros, and the author to prove quantitative nonlinear Poincaré inequalities and non-embedding theorems. Motivated by fundamental questions in geometric group theory and nonlinear functional analysis about superexpanders of logarithmic girth, our first main result is that every Ramanujan graph has long-range expansion. This gives explicit long-range expanders with logarithmic girth (such as Lubotzky--Phillips--Sarnak graphs) and establishes a strict hierarchy: Ramanujan expansion implies long-range expansion, which implies spectral expansion, with neither converse holding. We subsequently compare the different notions of expansion from the perspective of dynamics. Our second main result establishes cutoff with an explicit Gaussian limit profile for the random walk on every fixed-degree long-range expander sequence. In particular, the cutoff location and profile coincide with those for Ramanujan graphs. This offers intermediate progress between cutoff for Ramanujan graphs---proven by Lubetzky and Peres---and the long-standing conjecture of cutoff for transitive spectral expanders.

math.CO↗

arXiv2026

The threshold for online balancing of i.i.d. binary vectors

Consider the task of online vector balancing for stochastic arrivals $X_1,\ldots,{X_T}$, where the $X_i$ are independent uniformly random $d$--sparse binary vectors in $\{0,1\}^n$. This is a random analogue of the online Beck--Fiala problem. We show that uniformly for $2\le d\le n/2$ and $T = Θ(n)$, the optimal online prefix discrepancy $\max\limits_{t\leq T}\left\|\sum_{i=1}^tσ_i X_i\right\|_\infty$ is of order \[ Θ\big(\max\{\sqrt d,\log\log n\}\big). \] The upper bound is achieved by an efficient online algorithm. Thus, for $d\le(\log\log n)^2$, the optimal discrepancy is $Θ(\log\log n)$ and is independent of the sparsity up to constant factors, whereas above this scale it is $Θ(\sqrt d)$, matching the order of the offline discrepancy. This identifies the threshold at which sparsity begins to govern the online discrepancy of the random Beck--Fiala model.

math.PR↗

arXiv2026

Online Permutation Embedding: Optimal Stopping and Scaling Laws

We study optimal online algorithms for embedding a permutation $π$ of $[k]$ into an iid stream of uniform $[0,1]$ random variables. This problem is a broad generalization of the classical online monotone subsequence selection problem, recovered in the special case $π=\mathrm{Id}_k$. Our first contribution is an efficiently solvable dynamic program for the optimal embedding time of any $k$-permutation $π$. This dynamic program also yields an explicit optimal online embedding algorithm. We then investigate the asymptotic scaling of the optimal embedding time for uniformly random target permutations, as well as the extremal problem of identifying the permutations with largest expected online embedding time. Our second main result shows that, to first order, random permutations are strictly faster to embed than monotone permutations, which in turn are strictly faster to embed than the extremal permutations. This separation stands in sharp contrast to prevailing conjectures and heuristics in the offline theory of permutation embeddings.

math.PR↗

arXiv2026

Discrete Poincaré inequalities and universal approximators for random graphs

Nonlinear Poincaré inequalities are indispensable tools in the study of dimension reduction and low-distortion embeddings of graphs into metric spaces, and have found remarkable algorithmic applications. A basic open problem, posed by Jon Kleinberg (2013), asks whether the optimal nonlinear Poincaré constant for maps between two independent $3$-regular random graphs is dimension-free, i.e., independent of vertex-set sizes. We give a complete and affirmative resolution to Kleinberg's problem, also allowing for arbitrary graph degrees. As a corollary, we obtain a stochastic construction of $O(1)\text{-universal}$ approximators for random graphs, answering a question of Mendel and Naor.

math.MG↗

arXiv2026

Online Beck--Fiala Down to Logarithmic Sparsity

The Beck--Fiala conjecture asserts that every matrix $A\in\{0,1\}^{n\times T}$ with at most $d$ nonzero entries in each column has discrepancy $O(\sqrt d)$. A major breakthrough result of Bansal and Jiang recently established the validity of the conjecture for $d \ge \log(T)^2$. The present article extends the validity of the classical \textit{offline} Beck--Fiala conjecture to $d \ge \log(T)^{1+o(1)}$; moreover, the main thrust of the result is that it is actually obtained by an efficient \textit{online} algorithm that minimizes prefix discrepancy. The result is also essentially optimal, since online prefix discrepancy is known to scale as $ω(\sqrt{d})$ for $d =o(\log T)$. As an immediate corollary, the open question of online vector balancing in the Spencer setting is also resolved. The algorithm is based on a compactly supported Metropolis fixed-point walk, constructed by combining ideas from several recent works on the online Komlós problem. The proof was generated in conversation with ChatGPT 5.6 Pro; the authors provided high-level guidance in several rounds of prompting, followed by manual checking and rewriting of the proof.

math.CO↗

arXiv2026

A universal threshold for geometric embeddings of trees

A graph $G=(V,E)$ is geometrically embeddable into a normed space $X$ when there is a mapping $ζ: V\to X$ such that $\|ζ(v)-ζ(w)\|_X\leqslant 1$ if and only if $\{v,w\}\in E$, for all distinct $v,w\in V$. Our result is the following universal threshold for the embeddability of trees. Let $Δ\geqslant 3$, and let $N$ be sufficiently large in terms of $Δ$. Every $N$--vertex tree of maximal degree at most $Δ$ is embeddable into any normed space of dimension at least $64\,\frac{\log N}{\log\log N}$, and complete trees are non-embeddable into any normed space of dimension less than $\frac{1}{2}\,\frac{\log N}{\log\log N}$. In striking contrast, spectral expanders and random graphs are known to be non-embeddable in sublogarithmic dimension. Our result is based on a randomized embedding whose analysis utilizes the recent breakthroughs on Bourgain's slicing problem.

math.CO↗

arXiv2026

Metric Poincaré inequalities for graphs

This article obtains purely metric counterparts of cornerstone results in the theory of embedding graphs into normed spaces. Our first main result is a metric analogue of Matoušek's extrapolation relating the Poincaré constants $γ(G,\varrho^p)$ and $γ(G,\varrho^q)$ for any exponents $0 < p,q < \infty$, any bounded-degree expander graph $G$, and any target metric space $\mathcal{M}=(M,\varrho)$. Our second main result provides a sharp estimate of the Poincaré constant $γ(G,\varrho)$ in terms of the cardinalities of the vertex set of $G$ and the metric space $\mathcal{M}=(M,\varrho)$, in the setting of \textit{random} graphs. This yields optimal estimates on the minimum cardinality of (bi-Lipschitz) universal metric spaces for graphs, finally establishing a nonlinear analogue of Matoušek's celebrated "incompressibility" theorem (1996). Further, we obtain estimates on the nonlinear spectral gap of metric snowflakes and sharp lower bounds on the distortion of random regular graphs into arbitrary metric spaces. Our proofs develop new nonlinear techniques, including random compression methods and a novel structural dichotomy for metric embeddings.

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arXiv2025

A threshold for online balancing of sparse i.i.d. vectors

Consider the task of \textit{online} vector balancing for stochastic arrivals $(X_i)_{i \in [T]}$, where the time horizon satisfies $T = Θ(n)$, and the $X_i$ are i.i.d uniform $d$--sparse $n$--dimensional binary vectors, with $2\leq d \le (\log\log n)^2/\log\log\log n$. We show that for this range of parameters, every online algorithm incurs discrepancy at least $Ω(\log \log n)$, and there is an efficient algorithm which achieves a matching discrepancy bound of $O(\log\log n)$ w.h.p. This establishes an asymptotic gap, both existential and algorithmic, between the online and offline versions of the average--case Beck--Fiala problem. Strikingly, the optimal online discrepancy in the considered setting is order $\log \log n$, independent of $d$ and the norms of the vectors $(X_i)_i$. Our assumptions on $d$ are nearly optimal, as this independence ceases when $d=ω((\log\log n)^2)$.

math.PR↗

arXiv2025

Metric dimension reduction modulus for superlogarithmic distortion

The metric dimension reduction modulus $k^α_n(\ell_\infty)$ is the smallest $k$ such that every $n$--point metric space can be embedded into some $k$-dimensional normed space, with bi--Lipschitz distortion at most $α$. Determining sharp asymptotics for $k^α_n(\ell_\infty)$ is a fundamental task in metric geometry, with $α=Θ(\log n)$ bearing particular interest. A line of advances over the past decades has led to an upper bound on $k^α_n(\ell_\infty)$ for $α= Ω(\log n)$, but a matching lower bound has remained open. We close this gap, establishing: for every fixed $β> 0$, $$ k^α_n(\ell_\infty) =Θ\bigg(\frac{\log n}{\log(\fracα{\log n}+1)}\bigg)\quad \mbox{for every $α\geq β\log n$}. $$ This resolves a question from Naor's 2018 ICM plenary lecture. Our result is obtained by characterizing the minimum dimension $d$ for which, with high probability, a random regular graph admits an $α$--embedding into some $d$--dimensional normed space.

math.MG↗

arXiv2025

A combinatorial approach to nonlinear spectral gaps

A seminal open question of Pisier and Mendel--Naor asks whether every degree-regular graph which satisfies the classical discrete Poincaré inequality for scalar functions, also satisfies an analogous inequality for functions taking values in \textit{any} normed space with non-trivial cotype. Motivated by applications, it is also greatly important to quantify the dependence of the corresponding optimal Poincaré constant on the cotype $q$. Works of Odell--Schlumprecht (1994), Ozawa (2004), and Naor (2014) make substantial progress on the former question by providing a positive answer for normed spaces which also have an unconditional basis, in addition to finite cotype. However, little is known in the way of quantitative estimates: the mentioned results imply a bound on the Poincaré constant depending super-exponentially on $q$. We introduce a novel combinatorial framework for proving quantitative nonlinear spectral gap estimates. The centerpiece is a property of regular graphs that we call \emph{long range expansion}, which holds with high probability for random regular graphs. Our main result is that any regular graph with the long-range expansion property satisfies a discrete Poincaré inequality for any normed space with an unconditional basis and cotype $q$, with a Poincaré constant that depends \emph{polynomially} on $q$, which is optimal. As an application, any normed space with an unconditional basis which admits a low distortion embedding of an $n$-vertex random regular graph, must have cotype at least polylogarithmic in $n$. This extends a celebrated lower-bound of Matoušek for low distortion embeddings of random graphs into $\ell_q$ spaces.

math.MG↗

arXiv2025

Universal geometric non-embedding of random regular graphs

Let $Δ\ge 3$ be fixed, $n \ge n_Δ$ be a large integer. It is a classical result that $Δ$--regular expanders on $n$ vertices are not embeddable as geometric (distance) graphs into Euclidean space of dimension less than $c \log n$, for some universal constant $c$. We show that for typical $Δ$-regular graphs, this obstruction is universal with respect to the choice of norm. More precisely, for a uniform random $Δ$-regular graph $G$ on $n$ vertices, it holds with high probability: there is no normed space of dimension less than $c\log n$ which admits a geometric graph isomorphic to $G$. The proof is based on a seeded multiscale $\varepsilon$--net argument.

math.MG↗

arXiv2024

Locally seeded embeddings, and Ramsey numbers of bipartite graphs with sublinear bandwidth

A seminal result of Lee asserts that the Ramsey number of any bipartite $d$-degenerate graph $H$ satisfies $\log r(H) = \log n + O(d)$. In particular, this bound applies to every bipartite graph of maximal degree $Δ$. It remains a compelling challenge to identify conditions that guarantee that an $n$-vertex graph $H$ has Ramsey number linear in $n$, independently of $Δ$. Our contribution is a characterization of bipartite graphs with linear-size Ramsey numbers in terms of graph bandwidth, a notion of local connectivity. We prove that for any $n$-vertex bipartite graph $H$ with maximal degree at most $Δ$ and bandwidth $b(H)$ at most $\exp(-CΔ\logΔ)\,n$, we have $\log r(H) = \log n + O(1)$. This characterization is nearly optimal: for every $Δ$ there exists an $n$-vertex bipartite graph $H$ of degree at most $Δ$ and $b(H) \leq \exp(-cΔ)\,n$, such that $\log r(H) = \log n + Ω(Δ)$. We also provide bounds interpolating between these two bandwidth regimes.

math.CO↗

arXiv2024

Zero-One Laws for Random Feasibility Problems

We introduce a general random model of a combinatorial optimization problem with geometric structure that encapsulates both linear programming and integer linear programming. Let $Q$ be a bounded set called the feasible set, $E$ be an arbitrary set called the constraint set, and $A$ be a random linear transform. We define and study the $\ell^q$-margin, $M_q := d_q(AQ, E)$. The margin quantifies the feasibility of finding $y \in AQ$ satisfying the constraint $y \in E$. Our contribution is to establish strong concentration of the margin for any $q \in (2,\infty]$, assuming only that $E$ has permutation symmetry. The case of $q = \infty$ is of particular interest in applications -- specifically to combinatorial ``balancing'' problems -- and is markedly out of the reach of the classical isoperimetric and concentration-of-measure tools that suffice for $q \le 2$. Generality is a key feature of this result: we assume permutation symmetry of the constraint set and nothing else. This allows us to encode many optimization problems in terms of the margin, including random versions of: the closest vector problem, integer linear feasibility, perceptron-type problems, $\ell^q$-combinatorial discrepancy for $2 \le q \le \infty$, and matrix balancing. Concentration of the margin implies a host of new sharp threshold results in these models, and also greatly simplifies and extends some key known results.

math.PR↗

arXiv2024

On spectral outliers of inhomogeneous symmetric random matrices

Sharp conditions for the presence of spectral outliers are well understood for Wigner random matrices with iid entries. In the setting of inhomogeneous symmetric random matrices (i.e., matrices with a non-trivial variance profile), the corresponding problem has been considered only recently. Of special interest is the setting of sparse inhomogeneous matrices since sparsity is both a key feature and a technical obstacle in various aspects of random matrix theory. For such matrices, the largest of the variances of the entries has been used in the literature as a natural proxy for sparsity. We contribute sharp conditions in terms of this parameter for an inhomogeneous symmetric matrix with sub-Gaussian entries to have outliers. Our result implies a ``structural'' universality principle: the presence of outliers is only determined by the level of sparsity, rather than the detailed structure of the variance profile.

math.PR↗

arXiv2024

A note on the capacity of the binary perceptron

Determining the capacity $α_c$ of the Binary Perceptron is a long-standing problem. Krauth and Mezard (1989) conjectured an explicit value of $α_c$, approximately equal to .833, and a rigorous lower bound matching this prediction was recently established by Ding and Sun (2019). Regarding the upper bound, Kim and Roche (1998) and Talagrand (1999) independently showed that $α_c$ < .996, while Krauth and Mezard outlined an argument which can be used to show that $α_c$ < .847. The purpose of this expository note is to record a complete proof of the bound $α_c$ < .847. The proof is a conditional first moment method combined with known results on the spherical perceptron

math.PR↗

arXiv2023

Critical Window of The Symmetric Perceptron

We study the critical window of the symmetric binary perceptron, or equivalently, combinatorial discrepancy. Consider the problem of finding a binary vector $σ$ satisfying $\|Aσ\|_\infty \le K$, where $A$ is an $αn \times n$ matrix with iid Gaussian entries. For fixed $K$, at which densities $α$ is this constraint satisfaction problem (CSP) satisfiable? A sharp threshold was recently established by Perkins and Xu, and Abbe, Li, and Sly , answering this to first order. Namely, for each $K$ there exists an explicit critical density $α_c$ so that for any fixed $ε> 0$, with high probability the CSP is satisfiable for $αn < (α_c - ε) n$ and unsatisfiable for $αn > (α_c + ε) n$. This corresponds to a bound of $o(n)$ on the size of the critical window. We sharpen these results significantly, as well as provide exponential tail bounds. Our main result is that, perhaps surprisingly, the critical window is actually at most $O(\log n)$. More precisely, with high probability the CSP is satisfiable for $αn < α_c n -O(\log n)$ and unsatisfiable for any $αn > α_c n + ω(1)$. This implies the symmetric perceptron has nearly the "sharpest possible transition," adding it to a short list of CSP for which the critical window is rigorously known to be of near-constant width.

math.PR↗

arXiv2021

The Discrepancy of Random Rectangular Matrices

A recent approach to the Beck-Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models: matrices with Bernoulli or Poisson entries. For Poisson matrices, we further characterize the discrepancy for any rectangular aspect ratio. These results give sharp answers to questions of Hoberg and Rothvoss (SODA 2019) and Franks and Saks (Random Structures Algorithms 2020). Our main tool is a conditional second moment method combined with Stein's method of exchangeable pairs. While previous approaches are limited to dense matrices, our techniques allow us to work with matrices of all densities. This may be of independent interest for other sparse random constraint satisfaction problems.

math.PR↗

arXiv2012

The Zoo of Solitons for Curve Shortening in $\R^n$

We provide a detailed description of solutions of Curve Shortening in $\R^n$ that are invariant under some one-parameter symmetry group of the equation, paying particular attention to geometric properties of the curves, and the asymptotic properties of their ends. We find generalized helices, and a connection with curve shortening on the unit sphere $\Sph^{n-1}$. Expanding rotating solitons turn out to be asymptotic to generalized logarithmic spirals. In terms of asymptotic properties of their ends the rotating shrinking solitons are most complicated. We find that almost all of these solitons are asymptotic to circles. Many of the curve shortening solitons we discuss here are either space curves, or evolving space curves. In addition to the figures in this paper, we have prepared a number of animations of the solitons, which can be viewed at http://www.youtube.com/user/solitons2012/videos?view=1.

math.DG↗