arXiv subjects
Publications and source records attributed to Marcus Michelen.
Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification of all eventually non-negative polynomials. This generalizes a theorem of De Angelis, and proves a conjecture of Bergweiler, Eremenko and Sokal
We derive asymptotic formulas for the number of integer partitions with given sums of $j$th powers of the parts for $j$ belonging to a finite, non-empty set $J \subset \mathbb N$. The method we use is based on the `principle of maximum entropy' of Jaynes. This principle leads to an intuitive variational formula for the asymptotics of the logarithm of the number of constrained partitions as the solution to a convex optimization problem over real-valued functions.
Let $M_n$ be drawn uniformly from all $\pm 1$ symmetric $n \times n$ matrices. We show that the probability that $M_n$ is singular is at most $\exp(-c(n\log n)^{1/2})$, which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of $\exp(-c n^{1/2})$ on the singularity probability, our method is different and considerably simpler.
Let $f = \sum_{k=0}^n \varepsilon_k z^k$ be a random polynomial, where $\varepsilon_0,\ldots ,\varepsilon_n$ are iid standard Gaussian random variables, and let $ζ_1,\ldots,ζ_n$ denote the roots of $f$. We show that the point process determined by the magnitude of the roots $\{ 1-|ζ_1|,\ldots, 1-|ζ_n| \}$ tends to a Poisson point process at the scale $n^{-2}$ as $n\rightarrow \infty$. One consequence of this result is that it determines the magnitude of the closest root to the unit circle. In particular, we show that \[ \min_{k} ||ζ_k| - 1|n^2 \rightarrow \mathrm{Exp}(1/6),\] in distribution, where $\mathrm{Exp}(λ)$ denotes an exponential random variable of mean $λ^{-1}$. This resolves a conjecture of Shepp and Vanderbei from 1995 that was later studied by Konyagin and Schlag.
Let $f_n(z) = \sum_{k = 0}^n \varepsilon_k z^k$ be a random polynomial where $\varepsilon_0,\ldots,\varepsilon_n$ are i.i.d. random variables with $\mathbb{E} \varepsilon_1 = 0$ and $\mathbb{E} \varepsilon_1^2 = 1$. Letting $r_1, r_2,\ldots, r_k$ denote the real roots of $f_n$, we show that the point process defined by $\{|r_1| - 1,\ldots, |r_k| - 1 \}$ converges to a non-Poissonian limit on the scale of $n^{-1}$ as $n \to \infty$. Further, we show that for each $δ> 0$, $f_n$ has a real root within $Θ_δ(1/n)$ of the unit circle with probability at least $1 - δ$. This resolves a conjecture of Shepp and Vanderbei from 1995 by confirming its weakest form and refuting its strongest form.
An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than $n^{-.003}$, answering a question of Pittel. A lower bound of $n^{-1/2}$ was proven by Erdős and Richmond, and so this demonstrates that the probability decreases polynomially. Key to our argument is an asymptotic result of Pittel characterizing the joint distribution of the first rows and columns of a uniformly random partition, combined with a characterization of graphical partitions due to Erdős and Gallai. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.
We examine an interacting particle system on trees commonly referred to as the frog model. For its initial state, it begins with a single active particle at the root and i.i.d. $\mathrm{Poiss}(λ)$ many inactive particles at each non-root vertex. Active particles perform discrete time simple random walk and in the process activate any inactive particles they encounter. We show that for $\textit{every}$ non-amenable tree with bounded degree there exists a phase transition from transience to recurrence (with a non-trivial intermediate phase sometimes sandwiched in between) as $λ$ varies.
Let $X \in \{0,\ldots,n \}$ be a random variable, with mean $μ$ and standard deviation $σ$ and let \[f_X(z) = \sum_{k} \mathbb{P}(X = k) z^k, \] be its probability generating function. Pemantle conjectured that if $σ$ is large and $f_X$ has no roots close to $1\in \mathbb{C}$ then $X$ must be approximately normal. We completely resolve this conjecture in the following strong quantitative form, obtaining sharp bounds. If $δ= \min_ζ|ζ-1|$ over the complex roots $ζ$ of $f_X$, and $X^{\ast} := (X-μ)/σ$, then \[ \sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{\log n}{δσ} \right) \] where $Z \sim \mathcal{N}(0,1)$ is a standard normal. This gives the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. We also show that if $f_X$ has no roots with small argument, then $X$ must be approximately normal, again in a sharp quantitative form: if we set $δ= \min_ζ|\arg(ζ)|$ then \[ \sup_{t \in \mathbb{R}} \left|\mathbb{P}(X^{\ast} \leq t) - \mathbb{P}( Z \leq t) \, \right| = O\left(\frac{1}{δσ} \right). \] Using this result, we answer a question of Ghosh, Liggett and Pemantle by proving a sharp multivariate central limit theorem for random variables with real-stable probability generating functions.
We analyze the asymptotic behavior of the average maximal number of balls in a bin obtained by throwing uniformly at random $r$ balls without replacement into $n$ bins, $T$ times. Writing the expected maximum as $\frac{r}{n}T+ C_{n,r}\sqrt{T} + o(\sqrt{T})$, a recent preprint of Behrouzi-Far and Zeilberger asks for an explicit expression for $C_{n,r}$ in terms of $n,r$ and $π$. In this short note, we find an expression for $C_{n,r}$ in terms of $n, r$ and the expected maximum of $n$ independent standard Gaussians. This provides asymptotics for large $n$ as well as closed forms for small $n$---e.g. $C_{4,2} = \frac{3}{2 π^{3/2}} \arccos(-1/3)$---and shows that computing a closed form for $C_{n,r}$ is precisely as hard as the difficult question of finding the expected maximum of $n$ independent standard Gaussians.
We consider critical percolation on Galton-Watson trees and prove quenched analogues of classical theorems of critical branching processes. We show that the probability critical percolation reaches depth $n$ is asymptotic to a tree-dependent constant times $n^{-1}$. Similarly, conditioned on critical percolation reaching depth $n$, the number of vertices at depth $n$ in the critical percolation cluster almost surely converges in distribution to an exponential random variable with mean depending only on the offspring distribution. The incipient infinite cluster (IIC) is constructed for a.e. Galton-Watson tree and we prove a limit law for the number of vertices in the IIC at depth $n$, again depending only on the offspring distribution. Provided the offspring distribution used to generate these Galton-Watson trees has all finite moments, each of these results holds almost-surely.
We explore the survival function for percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results about the behavior of the random function $g(\mathbf{T} , \cdot)$, where $\mathbf{T}$ is drawn from the Galton-Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $k\text{th}$-order Taylor expansion of $g(\mathbf{T} , \cdot)$ at criticality in terms of limits of martingales defined from $\mathbf{T}$ (this requires a moment condition depending on $k$); and a proof that the $k\text{th}$ order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton-Watson tree.
For each $n$, let $X_n \in \{0,\ldots,n\}$ be a random variable with mean $μ_n$, standard deviation $σ_n$, and let \[ P_n(z) = \sum_{k=0}^n \mathbb{P}( X_n = k) z^k ,\] be its probability generating function. We show that if none of the complex zeros of the polynomials $\{ P_n(z)\}$ are contained in a neighbourhood of $1 \in \mathbb{C}$ and $σ_n > n^{\varepsilon}$ for some $\varepsilon >0$, then $ X_n^* =(X_n - μ_n)σ^{-1}_n$ tends to a normal random variable $Z \sim \mathcal{N}(0,1)$ in distribution as $n \rightarrow \infty$. Moreover, we show this result is sharp in the sense that there exist sequences of random variables $\{X_n\}$ with $σ_n > C\log n $ for which $P_n(z)$ has no roots near $1$ and $X_n^*$ is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of $P_n(z)$ and the distribution of the random variables $X_n$.
We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Additionally, we analyze the forward maximal weights along the backbone of the invasion cluster and prove a limit law for the process.