arXiv subjects

Renan Gross

Publications and source records attributed to Renan Gross.

15 recordsLinked to original sources

arXiv2026

From Kreweras walks to branching perimeter processes of percolated triangulations

In this note, using a result of Bernardi, Holden and Sun, we give an explicit geometric relation between the perimeter of the peeling process along the percolation interface of a triangulation, and the corresponding Kreweras walk. The relation naturally extends to the branching peeling exploration. This sheds light on the relation between the growth-fragmentation process and correlated Brownian excursions discovered by Da Silva, Powell and Watson.

math.CO↗

arXiv2026

Polynomial mixing for polygonal side matchings

We introduce a natural Markov chain on chord diagrams, which, at every step, selects two random chords and swaps them if doing so preserves the diagram's genus. This generalizes the chord swap chain on the Catalan structure of non-intersecting chord diagrams. We show that for fixed genus, the chain mixes in polynomial time.

math.PR↗

arXiv2026

Determinants of Laplacians on converging hyperbolic surfaces

Let $S_k$ be a sequence of compact hyperbolic surfaces of increasing volume which locally converges to a random rooted surface. We show that if the normalized sum of the reciprocal lengths of very short simple closed geodesics converges to 0, then the normalized logarithm of the determinant of the Laplacian of $S_k$ converges to a constant depending only the law of the limiting surface.

math.SP↗

arXiv2025

The distance problem via subadditivity

In a recent paper, Aldous, Blanc and Curien asked which distributions can be expressed as the distance between two independent random variables on some separable measured metric space. We show that every nonnegative discrete distribution whose support contains $0$ arises in this way, as well as a class of finitely supported distributions with density.

math.PR↗

arXiv2024

A sharp lower bound on the small eigenvalues of surfaces

Let $S$ be a compact hyperbolic surface of genus $g\geq 2$ and let $I(S) = \frac{1}{\mathrm{Vol}(S)}\int_{S} \frac{1}{\mathrm{Inj}(x)^2 \wedge 1} dx$, where $\mathrm{Inj}(x)$ is the injectivity radius at $x$. We prove that for any $k\in \{1,\ldots, 2g-3\}$, the $k$-th eigenvalue $\lambda_k$ of the Laplacian satisfies \begin{equation*} \lambda_k \geq \frac{c k^2}{I(S) g^2} \, , \end{equation*} where $c>0$ is some universal constant. We use this bound to prove the heat kernel estimate \begin{equation*} \frac{1}{\mathrm{Vol}(S)} \int_S \Big| p_t(x,x) -\frac{1}{\mathrm{Vol}(S)} \Big | ~dx \leq C \sqrt{ \frac{I(S)}{t}} \qquad \forall t \geq 1 \, , \end{equation*} where $C<\infty$ is some universal constant. These bounds are optimal in the sense that for every $g\geq 2$ there exists a compact hyperbolic surface of genus $g$ satisfying the reverse inequalities with different constants.

math.SP↗

arXiv2023

Randomly twisted hypercubes -- between structure and randomness

Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these graphs have optimal diameter. We study twisted hypercubes in the setting where the instances can have general dependence, and also in the particular case where they are identical. We show that the resultant graph shares properties with random regular graphs, including small diameter, large vertex expansion, a semicircle law for its eigenvalues and no non-trivial automorphisms. However, in contrast to random regular graphs, twisted hypercubes allow for short routing schemes.

math.CO↗

arXiv2022

Noise sensitivity from fractional query algorithms and the axis-aligned Laplacian

We introduce the notion of classical fractional query algorithms, which generalize decision trees in the average-case setting, and can potentially perform better than them. We show that the limiting run-time complexity of a natural class of these algorithms obeys the non-linear partial differential equation $\min_{k}\partial^{2}u/\partial x_{k}^{2}=-2$, and that the individual bit revealment satisfies the Schramm-Steif bound for Fourier weight, connecting noise sensitivity with PDEs. We discuss relations with other decision tree results.

cs.CC↗

arXiv2020

Brownian motion can feel the shape of a drum

We study the scenery reconstruction problem on the $d$-dimensional torus, proving that a criterion on Fourier coefficients obtained by Matzinger and Lember (2006) for discrete cycles applies also in continuous spaces. In particular, with the right drift, Brownian motion can be used to reconstruct any scenery. To this end, we prove an injectivity property of an infinite Vandermonde matrix.

math.PR↗

arXiv2020

Concentration on the Boolean hypercube via pathwise stochastic analysis

We develop a new technique for proving concentration inequalities which relate between the variance and influences of Boolean functions. Using this technique, we 1. Settle a conjecture of Talagrand [Tal97] proving that $$\int_{\left\{ -1,1\right\} ^{n}}\sqrt{h_{f}\left(x\right)}dμ\geq C\cdot\mathrm{var}\left(f\right)\cdot\left(\log\left(\frac{1}{\sum\mathrm{Inf}_{i}^{2}\left(f\right)}\right)\right)^{1/2},$$ where $h_{f}\left(x\right)$ is the number of edges at $x$ along which $f$ changes its value, and $\mathrm{Inf}_{i}\left(f\right)$ is the influence of the $i$-th coordinate. 2. Strengthen several classical inequalities concerning the influences of a Boolean function, showing that near-maximizers must have large vertex boundaries. An inequality due to Talagrand states that for a Boolean function $f$, $\mathrm{var}\left(f\right)\leq C\sum_{i=1}^{n}\frac{\mathrm{Inf}_{i}\left(f\right)}{1+\log\left(1/\mathrm{Inf}_{i}\left(f\right)\right)}$. We give a lower bound for the size of the vertex boundary of functions saturating this inequality. As a corollary, we show that for sets that satisfy the edge-isoperimetric inequality or the Kahn-Kalai-Linial inequality up to a constant, a constant proportion of the mass is in the inner vertex boundary. 3. Improve a quantitative relation between influences and noise stability given by Keller and Kindler. Our proofs rely on techniques based on stochastic calculus, and bypass the use of hypercontractivity common to previous proofs.

math.PR↗

arXiv2019

A phase diagram for bacterial swarming

Bacterial swarming is a rapid mass-migration, in which thousands of cells spread collectively to colonize a surface. Physically, swarming is a natural example of active particles that use energy to generate motion. Accordingly, understanding the constraints physics imposes on the dynamics is essential to understand the mechanisms underlying the swarming phenomenon. We present new experiments of swarming Bacillus subtilis mutants with different aspect ratios and densities. Analyzing the dynamics reveals a rich phase diagram of qualitatively distinct swarming regimes, describing how the shape and density of cells govern the global dynamical characteristics of the entire swarm. Moreover, we show that under standard conditions bacteria inhabit a region of phase space that is associated with rapid mixing and robust dynamics, with homogeneous density and no preferred direction of motion. This contrasts characteristic clustering behavior of self-propelled rods that is recovered only for very elongated mutant species. Thus, bacteria have adapted their physics to optimize the principle functions assumed for swarming.

cond-mat.soft↗

arXiv2019

A conformal Skorokhod embedding

Start a planar Brownian motion and let it run until it hits some given barrier. We show that the barrier may be crafted so that the x coordinate at the hitting time has any prescribed centered distribution with finite variance. This provides a new, complex-analytic proof of the Skorokhod embedding theorem. Our method is constructive and can give an explicit description of the barrier.

math.PR↗

arXiv2018

Exponential random graphs behave like mixtures of stochastic block models

We study the behavior of exponential random graphs in both the sparse and the dense regime. We show that exponential random graphs are approximate mixtures of graphs with independent edges whose probability matrices are critical points of an associated functional, thereby satisfying a certain matrix equation. In the dense regime, every solution to this equation is close to a block matrix, concluding that the exponential random graph behaves roughly like a mixture of stochastic block models. We also show existence and uniqueness of solutions to this equation for several families of exponential random graphs, including the case where the subgraphs are counted with positive weights and the case where all weights are small in absolute value. In particular, this generalizes some of the results in a paper by Chatterjee and Diaconis from the dense regime to the sparse regime and strengthens their bounds from the cut-metric to the one-metric.

math.PR↗

arXiv2018

Decomposition of mean-field Gibbs distributions into product measures

We show that under a low complexity condition on the gradient of a Hamiltonian, Gibbs distributions on the Boolean hypercube are approximate mixtures of product measures whose probability vectors are critical points of an associated mean-field functional. This extends a previous work by the first author. As an application, we demonstrate how this framework helps characterize both Ising models satisfying a mean-field condition and the conditional distributions which arise in the emerging theory of nonlinear large deviations, both in the dense case and in the polynomially-sparse case.

math.PR↗

arXiv2017

Indistinguishable sceneries on the Boolean hypercube

We show that the scenery reconstruction problem on the Boolean hypercube is in general impossible. This is done by using locally biased functions, in which every vertex has a constant fraction of neighbors colored by $1$, and locally stable functions, in which every vertex has a constant fraction of neighbors colored by its own color. Our methods are constructive, and also give super-polynomial lower bounds on the number of locally biased and locally stable functions. We further show similar results for $\mathbb{Z}^n$ and other graphs, and offer several follow-up questions.

math.CO↗

arXiv2014

Bounding the seed length of Miller and Shi's unbounded randomness expansion protocol

Recent randomness expansion protocols have been proposed which are able to generate an unbounded amount of randomness from a finite amount of truly random initial seed. One such protocol, given by Miller and Shi, uses a pair of non-signaling untrusted quantum mechanical devices. These play XOR games with inputs given by the user in order to generate an output. Here we present an analysis of the required seed size, giving explicit upper bounds for the number of initial random bits needed to jump-start the protocol. The bits output from such a protocol are $\varepsilon$-close to uniform even against quantum adversaries. Our analysis yields that for a statistical distance of $\varepsilon=10^{-1}$ and $\varepsilon=10^{-6}$ from uniformity, the number of required bits is smaller than 225,000 and 715,000, respectively; in general it grows as $O(\log\frac{1}{\varepsilon})$.

quant-ph↗