Searcharxiv⌕ Search

arXiv subjects

Marcus Michelen

Publications and source records attributed to Marcus Michelen.

At least 37 records · Page 2Linked to original sources
arXiv2024

The binomial random graph is a bad inducer

For a finite graph $F$ and a value $p \in [0,1]$, let $I(F,p)$ denote the largest $y$ for which there is a sequence of graphs of edge density approaching $p$ so that the induced $F$-density of the sequence approaches $y$. We show that for all $F$ on at least three vertices and all $p \in (0,1)$, the binomial random graph $G(n,p)$ has induced $F$-density strictly less than $I(F,p).$ This provides a negative answer to a problem posed by Liu, Mubayi and Reiher. Our approach is in the limiting setting of graphons, and we in fact show a stronger result: the binomial random graph is never a \emph{local} maximum in the space of graphons of edge density $p$. This is done by finding a sequence of balanced perturbations of arbitrarily small norm that increase the $F$-density.

math.CO↗
arXiv2024

Realizability of hypergraphs and high-dimensional contingency tables with random degrees and marginals

A result of Deza, Levin, Meesum, and Onn shows that the problem of deciding if a given sequence is the degree sequence of a 3-uniform hypergraph is NP complete. We tackle this problem in the random case and show that a random integer partition can be realized as the degree sequence of a $3$-uniform hypergraph with high probability. These results are in stark contrast with the case of graphs, where a classical result of Erdős and Gallai provides an efficient algorithm for checking if a sequence is a degree sequence of a graph and a result of Pittel shows that with high probability a random partition is not the degree sequence of a graph. By the same method, we address analogous realizability problems about high-dimensional binary contingency tables. We prove that if $(λ,μ,ν)$ are three independent random partitions then with high probability one can construct a three-dimensional binary contingency table with marginals $(λ,μ,ν)$. Conversely, if one insists that the contingency table forms a pyramid shape, then we show that with high probability one cannot construct such a contingency table. These two results confirm two conjectures of Pak and Panova.

math.CO↗
arXiv2024

On random polynomials with an intermediate number of real roots

For each $α\in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{α+ o(1)}$ with probability tending to one as the degree $n$ tends to infinity, where $(\varepsilon_k)$ is a sequence of i.i.d. (real) random variables of finite mean satisfying a mild anti-concentration assumption. In particular, this includes the case when $(\varepsilon_k)$ is a sequence of i.i.d. standard Gaussian or Rademacher random variables. This result confirms a conjecture of O. Nguyen from 2019. More generally, our main results also describe several statistical properties for the number of real roots of $f_n$, including the asymptotic behavior of the variance and a central limit theorem.

math.PR↗
arXiv2024

Sums of random polynomials with differing degrees

Let $μ$ and $ν$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$, whose roots are chosen independently from $μ$ and $ν$, respectively. Under assumptions on the measures $μ$ and $ν$, the limiting distribution for the zeros of the sum $p+q$ was by computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021) 124719] as $n \to \infty$. In this paper, we generalize and extend this result to the case where $p$ and $q$ have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of $μ$ and $ν$, scaled by the limiting ratio of the degrees of $p$ and $q$. Additionally, our approach provides a complete description of the limiting distribution for the zeros of $p + q$ for any pair of measures $μ$ and $ν$, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.

math.PR↗
arXiv2024

The frog model on Galton-Watson trees

We consider an interacting particle system on trees known as the frog model: initially, a single active particle begins at the root and i.i.d.~$\mathrm{Poiss}(λ)$ many inactive particles are placed at each non-root vertex. Active particles perform discrete time simple random walk and activate the inactive particles they encounter. We show that for Galton-Watson trees with offspring distributions $Z$ satisfying $\mathbf{P}(Z \geq 2) = 1$ and $\mathbf{E}[Z^{4 + ε}] < \infty$ for some $ε> 0$, there is a critical value $λ_c\in(0,\infty)$ separating recurrent and transient regimes for almost surely every tree, thereby answering a question of Hoffman-Johnson-Junge. In addition, we also establish that this critical parameter depends on the entire offspring distribution, not just the maximum value of $Z$, answering another question of Hoffman-Johnson-Junge and showing that the frog model and contact process behave differently on Galton-Watson trees.

math.PR↗
arXiv2023

A new lower bound for sphere packing

We show there exists a packing of identical spheres in $\mathbb{R}^d$ with density at least \[ (1-o(1))\frac{d \log d}{2^{d+1}}\, , \] as $d\to\infty$. This improves upon previous bounds for general $d$ by a factor of order $\log d$ and is the first asymptotically growing improvement to Rogers' bound from 1947.

math.MG↗
arXiv2023

The least singular value of a random symmetric matrix

Let $A$ be a $n \times n$ symmetric matrix with $(A_{i,j})_{i\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\mathbb{P}(σ_{\min}(A) \leq \varepsilon/\sqrt{n}) \leq C \varepsilon + e^{-cn},$$ where $σ_{\min}(A)$ denotes the least singular value of $A$ and the constants $C,c>0 $ depend only on the distribution of the entries of $A$. This result confirms a folklore conjecture on the lower-tail asymptotics of the least singular value of random symmetric matrices and is best possible up to the dependence of the constants on the distribution of $A_{i,j}$. Along the way, we prove that the probability $A$ has a repeated eigenvalue is $e^{-Ω(n)}$, thus confirming a conjecture of Nguyen, Tao and Vu.

math.PR↗
arXiv2023

Almost sure behavior of the zeros of iterated derivatives of random polynomials

Let $Z_1,\, Z_2,\dots$ be independent and identically distributed complex random variables with common distribution $μ$ and set $$ P_n(z) := (z - Z_1)\cdots (z - Z_n)\,. $$ Recently, Angst, Malicet and Poly proved that the critical points of $P_n$ converge in an almost-sure sense to the measure $μ$ as $n$ tends to infinity, thereby confirming a conjecture of Cheung-Ng-Yam and Kabluchko. In this short note, we prove for any fixed $k\in \mathbb{N}$, the empirical measure of zeros of the $k$th derivative of $P_n$ converges to $μ$ in the almost sure sense, as conjectured by Angst-Malicet-Poly.

math.PR↗
arXiv2023

Quasipolynomial-time algorithms for Gibbs point processes

We demonstrate a quasipolynomial-time deterministic approximation algorithm for the partition function of a Gibbs point process interacting via a finite-range stable potential. This result holds for all activities $λ$ for which the partition function satisfies a zero-free assumption in a neighborhood of the interval $[0,λ]$. As a corollary, for all finite-range stable potentials we obtain a quasipolynomial-time determinsitic algorithm for all $λ< /(e^{B + 1} \hat C_ϕ)$ where $\hat C_ϕ$ is a temperedness parameter and $B$ is the stability constant of $ϕ$. In the special case of a repulsive potential such as the hard-sphere gas we improve the range of activity by a factor of at least $e^2$ and obtain a quasipolynomial-time deterministic approximation algorithm for all $λ< e/Δ_ϕ$, where $Δ_ϕ$ is the potential-weighted connective constant of the potential $ϕ$. Our algorithm approximates coefficients of the cluster expansion of the partition function and uses the interpolation method of Barvinok to extend this approximation throughout the zero-free region.

cs.DS↗
arXiv2023

Optimal mixing of the down-up walk on independent sets of a given size

Let $G$ be a graph on $n$ vertices of maximum degree $Δ$. We show that, for any $δ> 0$, the down-up walk on independent sets of size $k \leq (1-δ)α_c(Δ)n$ mixes in time $O_{Δ,δ}(k\log{n})$, thereby resolving a conjecture of Davies and Perkins in an optimal form. Here, $α_{c}(Δ)n$ is the NP-hardness threshold for the problem of counting independent sets of a given size in a graph on $n$ vertices of maximum degree $Δ$. Our mixing time has optimal dependence on $k,n$ for the entire range of $k$; previously, even polynomial mixing was not known. In fact, for $k = Ω_Δ(n)$ in this range, we establish a log-Sobolev inequality with optimal constant $Ω_{Δ,δ}(1/n)$. At the heart of our proof are three new ingredients, which may be of independent interest. The first is a method for lifting $\ell_\infty$-independence from a suitable distribution on the discrete cube -- in this case, the hard-core model -- to the slice by proving stability of an Edgeworth expansion using a multivariate zero-free region for the base distribution. The second is a generalization of the Lee-Yau induction to prove log-Sobolev inequalities for distributions on the slice with considerably less symmetry than the uniform distribution. The third is a sharp decomposition-type result which provides a lossless comparison between the Dirichlet form of the original Markov chain and that of the so-called projected chain in the presence of a contractive coupling.

cs.DS↗
arXiv2022

Zeros of a growing number of derivatives of random polynomials with independent roots

Let $X_1,X_2,\ldots$ be independent and identically distributed random variables in $\mathbb{C}$ chosen from a probability measure $μ$ and define the random polynomial $$ P_n(z)=(z-X_1)\ldots(z-X_n)\,. $$ We show that for any sequence $k = k(n)$ satisfying $k \leq \log n / (5 \log\log n)$, the zeros of the $k$th derivative of $P_n$ are asymptotically distributed according to the same measure $μ$. This extends work of Kabluchko, which proved the $k = 1$ case, as well as Byun, Lee and Reddy who proved the fixed $k$ case.

math.PR↗
arXiv2022

Analyticity for classical gasses via recursion

We give a new criterion for a classical gas with a repulsive pair potential to exhibit uniqueness of the infinite volume Gibbs measure and analyticity of the pressure. Our improvement on the bound for analyticity is by a factor $e^2$ over the classical cluster expansion approach and a factor $e$ over the known limit of cluster expansion convergence. The criterion is based on a contractive property of a recursive computation of the density of a point process. The key ingredients in our proofs include an integral identity for the density of a Gibbs point process and an adaptation of the algorithmic correlation decay method from theoretical computer science. We also deduce from our results an improved bound for analyticity of the pressure as a function of the density.

math-ph↗
arXiv2022

Anti-concentration of random variables from zero-free regions

This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let $X$ be a random variable taking values in $\{0,\ldots,n\}$ with $\mathbb{P}(X = 0)\mathbb{P}(X = n) > 0$ and with probability generating function $f_X$. We show that if all of the zeros $ζ$ of $f_X$ satisfy $|\arg(ζ)| \geq δ$ and $R^{-1} \leq |ζ| \leq R$ then \[ \operatorname{Var}(X) \geq c R^{-2π/δ}n, \] where $c > 0$ is a absolute constant. We show that this result is sharp, up to the factor $2$ in the exponent of $R$. As a consequence, we are able to deduce a Littlewood--Offord type theorem for random variables that are not necessarily sums of i.i.d.\ random variables.

math.PR↗
arXiv2022

Strong spatial mixing for repulsive point processes

We prove that a Gibbs point process interacting via a finite-range, repulsive potential $ϕ$ exhibits a strong spatial mixing property for activities $λ< e/Δ_ϕ$, where $Δ_ϕ$ is the potential-weighted connective constant of $ϕ$, defined recently in [MP21]. Using this we derive several analytic and algorithmic consequences when $λ$ satisfies this bound: (1) We prove new identities for the infinite volume pressure and surface pressure of such a process (and in the case of the surface pressure establish its existence). (2) We prove that local block dynamics for sampling from the model on a box of volume $N$ in $\mathbb R^d$ mixes in time $O(N \log N)$, giving efficient randomized algorithms to approximate the partition function and approximately sample from these models. (3) We use the above identities and algorithms to give efficient approximation algorithms for the pressure and surface pressure.

math.PR↗
arXiv2022

Direct design of biquad filter cascades with deep learning by sampling random polynomials

Designing infinite impulse response filters to match an arbitrary magnitude response requires specialized techniques. Methods like modified Yule-Walker are relatively efficient, but may not be sufficiently accurate in matching high order responses. On the other hand, iterative optimization techniques often enable superior performance, but come at the cost of longer run-times and are sensitive to initial conditions, requiring manual tuning. In this work, we address some of these limitations by learning a direct mapping from the target magnitude response to the filter coefficient space with a neural network trained on millions of random filters. We demonstrate our approach enables both fast and accurate estimation of filter coefficients given a desired response. We investigate training with different families of random filters, and find training with a variety of filter families enables better generalization when estimating real-world filters, using head-related transfer functions and guitar cabinets as case studies. We compare our method against existing methods including modified Yule-Walker and gradient descent and show our approach is, on average, both faster and more accurate.

eess.SP↗
arXiv2021

Potential-weighted connective constants and uniqueness of Gibbs measures

We define a potential-weighted connective constant that measures the effective strength of a repulsive pair potential of a Gibbs point process modulated by the geometry of the underlying space. We then show that this definition leads to improved bounds for Gibbs uniqueness for all non-trivial repulsive pair potentials on $\mathbb R^d$ and other metric measure spaces. We do this by constructing a tree-branching collection of densities associated to the point process that captures the interplay between the potential and the geometry of the space. When the activity is small as a function of the potential-weighted connective constant this object exhibits an infinite volume uniqueness property. On the other hand, we show that our uniqueness bound can be tight for certain spaces: the same infinite volume object exhibits non-uniqueness for activities above our bound in the case when the underlying space has the geometry of a tree.

math.PR↗
arXiv2021

Success probability for selectively neutral invading species in the line model with a random fitness landscape

We consider a spatial (line) model for invasion of a population by a single mutant with a stochastically selectively neutral fitness landscape, independent from the fitness landscape for non-mutants. This model is similar to those considered in Farhang-Sardroodi et al. [PLOS Comput. Biol., 13(11), 2017; J. R. Soc. Interface, 16(157), 2019]. We show that the probability of mutant fixation in a population of size $N$, starting from a single mutant, is greater than $1/N$, which would be the case if there were no variation in fitness whatsoever. In the small variation regime, we recover precise asymptotics for the success probability of the mutant. This demonstrates that the introduction of randomness provides an advantage to minority mutations in this model, and shows that the advantage increases with the system size. We further demonstrate that the mutants have an advantage in this setting only because they are better at exploiting unusually favorable environments when they arise, and not because they are any better at exploiting pockets of favorability in an environment that is selectively neutral overall.

math.PR↗