arXiv subjects
Publications and source records attributed to Marcus Michelen.
The hard sphere model is a classical model from statistical physics in which particles are represented by equal-sized spheres. Longstanding predictions from the physics literature indicate that in $\mathbb{R}^2$ and $\mathbb{R}^3$ the system undergoes a phase transition, but it remains a major open problem to confirm this. We prove the existence of a phase transition for this model in the hyperbolic plane.
A Gibbs point process models particles interacting in the continuum through a potential. Among the most classical examples is the hard-sphere model, where given an activity parameter $\lambda$, a radius $r$, and a bounded set $\Lambda \subset \mathbb{R}^d$ one samples a Poisson process of intensity $\lambda$ in $\Lambda$ conditioned on the points forming the centers of an $r$-sphere packing. We prove uniqueness of infinite-volume Gibbs measure, analyticity of the pressure, and various notions of spatial and temporal mixing for activities up to what we define as the spectral threshold $\lambda_{spec}$ of the potential. For each fixed dimension $d \geq 2$, this improves the uniqueness and analyticity bounds for the hard-sphere model. As $d \to \infty$, our improvement over the classical bounds grows exponentially. We also prove an optimal mixing time bound for heat bath dynamics for the hard-sphere model up to an expected density of $\Theta(d / 2^d)$, the first result that asymptotically matches the maximum density for rapid mixing predicted by Parisi and Zamponi. We also exhibit repulsive, radial pair potentials for which $\lambda_{spec} = + \infty$, showing that the corresponding Gibbs point processes have no phase transition at any activity $\lambda > 0$. Further, in dimensions $8$ and $24$ we exhibit such a potential with no phase transition for which the work of Cohn-Kumar-Miller-Radchenko-Viazovska proves that the unique ground state at any fixed density is given by the $E_8$ and Leech lattices, respectively. Our work builds upon a 2013 work of Kondratiev-Kuna-Ohlerich that implicitly defined $\lambda_{spec}$ and proved a spectral gap for a Glauber-like continuum birth-death dynamics. Our main work shows that such a spectral gap implies several strong notions of absence of phase transition and analyzes the behavior of $\lambda_{spec}$ for interesting potentials.
A recent breakthrough of Chen, Chen, Chen, Yin, and Zhang shows rapid mixing for Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. Their approach builds upon the literature of various local-to-global techniques and applies to a more general setting of discrete distributions supported on downward-closed set families. We give a short and self-contained proof via a Bochner--Bakry--\'{E}mery approach and directly show a Poincar\'e inequality by expanding the Dirichlet form in terms of the $L^2$-norm of the generator applied to a test function and eliminating a sum of squares term. Our proof is a streamlined version of an argument of Kondratiev, Kuna, and Ohlerich used to study spatial birth-and-death dynamics for Gibbs point processes in the continuum, which we adapt to the discrete setting.
A Littlewood polynomial is a polynomial whose coefficients lie in $\{- 1, +1\}$. While the majority of roots of a Littlewood polynomial of large degree are near the unit circle, numerical experiments suggest that when plotting the roots of \emph{all} Littlewood polynomials of a given large degree, striking fractal structures appear away from the unit circle. These fractals resemble the attractor of a certain iterated function system and are known as \emph{dragon curves}. In this note, we provide a rigorous explanation of this phenomenon, along with an analysis of a random variant, saying that such fractal behavior is typical.
Consider a random geometric graph $G_M(n;r)$ whose vertex set consists of $n$ points chosen independently and uniformly from a Riemannian manifold $M$, with edges joining pairs of vertices whose distance in the metric $d_M$ is at most $r$. Let $\Delta$ denote the expected average degree of the graph. As is the case for Erd\H{o}s-R\'enyi graphs, there is a critical value $\Delta_c$, depending only on the dimension of $M$, such that if $\Delta > \Delta_c$ then $G_M(n;r)$ has a giant component. We show that whenever $\Delta > \Delta_c$, the giant component of $G_M(n;r)$, equipped with the graph distance, converges to the underlying manifold $M$ in the Gromov-Hausdorff distance after rescaling by an appropriate deterministic factor. Our result holds for $\Delta$ depending on $n$ as well, provided $\Delta = o(n)$ and $\Delta \geq \Delta_c + \varepsilon$ for any fixed $\varepsilon > 0$. As a consequence, we show that for any pair of non-isometric compact Riemannian manifolds $M_1$ and $M_2$, there is a polynomial-time algorithm that distinguishes random geometric graphs on $M_1$ and $M_2$ throughout this regime of $\Delta.$ In the thermodynamic regime -- i.e.\ when $\Delta$ is constant -- our results appear to be new even in the classical cases where $M$ is a sphere or a torus. Our proof makes use of techniques from first-passage percolation which allow us to understand the long-range behavior of the graph distance on small, approximately Euclidean patches of $M$, together with global arguments that glue these local estimates into a global description.
For $r \ge 2$ and a graph $G$, let $α_{r}(G)$ be the maximum number of vertices in a $K_r$-free subgraph of $G$. We investigate the value $α_{r}(G)$ when $G$ is the random graph $G \sim G_{n, 1/2}$ and discover the following phenomenon: with high probability, $α_r(G)$ lies in an interval of constant length that varies in a non-monotonic fashion from $1$ to $\lfloor r/2\rfloor+1$ depending on the value of $n$. The special case $r=2$ corresponds to the independence number of random graphs which is well-known to have two-point concentration; our results therefore extend and generalize this basic fact in random graph theory, showing more complicated behavior when $r>2$. We also prove similar results where $K_r$ is replaced by any color critical graph like $C_5$.
The (strong) Bruhat order for permutations provides a partial ordering defined as follows: two permutations are comparable if one can be obtained from the other by a sequence of adjacent transpositions that each increase the number of inversions by $1$. Given two random permutations, what is the probability that they are comparable in the Bruhat order? This problem was first considered in a 2006 work of Hammett and Pittel, which showed an exponential lower bound and a polynomial upper bound. The lower bound was very recently improved to the subexponential bound of $\exp(-n^{1/2 + o(1)})$ by Boretsky, Cornejo, Hodges, Horn, Lesnevich, and McAllister. Hammett and Pittel predicted that the probability should decrease polynomially. We show that the probability decreases faster than any polynomial and is on the order of $\exp(-Θ(\log^2 n))$.
Assume that $n = 2k$ potential roommates each have an ordered preference of the $n-1$ others. A stable matching is a perfect matching of the $n$ roommates in which no two unmatched people prefer each other to their matched partners. In their seminal 1962 stable marriage paper, Gale and Shapley noted that not every instance of the stable roommates problem admits a stable matching. In the case when the preferences are chosen uniformly at random, Gusfield and Irving predicted in 1989 that there is no stable matching with high probability for large $n$. We prove this conjecture and show that for $n$ sufficiently large, the probability there is a stable matching is at most $n^{-1/17}$.
Let $f_n$ be a random polynomial of degree $n$ with i.i.d. mean-zero and finite variance random coefficients. It is well known that the roots of $f_n$ cluster uniformly around the unit circle as $n$ grows large. We give a simple and self-contained proof of local universality for the correlation functions of the roots at the microscopic scale $1/n$ around a fixed point on the circle. While previous proofs of local universality were focused on studying the logarithmic potential of $f_n$, we instead directly compare the scaled random polynomial to a limiting Gaussian analytic function, and establish convergence of correlations via a soft argument, using only basic complex analysis and an anti-concentration bound of Esseen.
We give a polynomial improvement to the cycle-complete Ramsey numbers \[ r(C_{\ell},K_k) \geq k^{1+1/(\ell- 2) + \varepsilon_{\ell} + o(1)}, \] for all fixed odd $\ell > 7$ with $k \rightarrow \infty$, for some $\varepsilon_{\ell} > 0$.
Consider a random power series of the form $P(z) = \sum_{n\ge 1} \varepsilon_n a_n z^{n}$ where $a_n \in \mathbb{C}$ are deterministic and $\varepsilon_n$ are chosen independently and uniformly at random from $\{\pm 1\}$. Kolmogorov's three-series theorem states that if $\sum_{n} |a_n|^2 = \infty$ then $P(z)$ almost-surely diverges at almost every $z$ with $|z| = 1$. Dvoretzky and Erdős proved in 1959 that if $|a_n| = Ω(1/\sqrt{n})$ then in fact $P$ almost surely diverges at every $|z| = 1$. Erdős then asked in 1961 if this is sharp, meaning that if $|a_n| = o(1/\sqrt{n})$ then there is almost surely some convergent point $z$ with $|z| = 1$. We prove this in a strong sense and show that if $a_n = o(1/\sqrt{n})$ then in fact the set of convergent points of $P$ with $|z| = 1$ has Hausdorff dimension $1$.
Let $f_n$ be a random polynomial of degree $n$, whose coefficients are independent and identically distributed random variables with mean-zero and variance one. Let $Δ(f_n)$ denote the discriminant of $f_n$, that is $Δ(f_n) = A^{2n-2}\prod_{i < j} (α_j - α_i)^2$ where $A$ is the leading coefficient of $f_n$ and $α_1,\ldotsα_n$ are its roots. We prove that with high probability $$|Δ(f_n)| = n^{2n} e^{-{\sf D}_\ast n(1+o(1))}$$ as $n\to \infty$, for some explicit universal constant ${\sf D}_\ast>0$. A key step in the proof is an analytic representation for the logarithm of the discriminant, which captures both the distributional reciprocal symmetry of the random roots and the cancellations this symmetry induces.
We prove a new lower bound for the off-diagonal Ramsey numbers, \[ R(3,k) \geq \bigg( \frac{1}{3}+ o(1) \bigg) \frac{k^2}{\log k }\, , \] thereby narrowing the gap between the upper and lower bounds to a factor of $3+o(1)$. This improves the best known lower bound of $(1/4+o(1))k^2/\log k$ due, independently, to Bohman and Keevash, and Fiz Pontiveros, Griffiths and Morris, resulting from their celebrated analysis of the triangle-free process. As a consequence, we disprove a conjecture of Fiz Pontiveros, Griffiths and Morris that the constant $1/4$ is sharp.
Let $f_n$ be a random polynomial of degree $n\ge 2$ whose coefficients are independent and identically distributed random variables. We study the separation distances between roots of $f_n$ and prove that the set of these distances, normalized by $n^{-5/4}$, converges in distribution as $n\to \infty$ to a non-homogeneous Poisson point process. As a corollary, we deduce that the minimal separation distance between roots of $f_n$, normalized by $n^{-5/4}$ has a non-trivial limit law. In the course of the proof, we establish a related result which may be of independent interest: a Taylor series with random i.i.d. coefficients almost-surely does not have a double zero anywhere other than the origin.
We develop a theory of random flat-foldable origami. Given a crease pattern, we consider a uniformly random assignment of mountain and valley creases, conditioned on the assignment being flat-foldable at each vertex. A natural method to approximately sample from this distribution is via the face-flip Markov chain where one selects a face of the crease pattern uniformly at random and, if possible, flips all edges of that face from mountain to valley and vice-versa. We prove that this chain mixes rapidly for several natural families of origami tessellations -- the square twist, the square grid, and the Miura-ori -- as well as for the single-vertex crease pattern. We also compare local to global flat-foldability and show that on the square grid, a random locally flat-foldable configuration is exponentially unlikely to be globally flat-foldable.
Smale's Seventh Problem asks for an efficient algorithm to generate a configuration of $n$ points on the sphere that nearly minimizes the logarithmic energy. As a candidate starting configuration for this problem, Armentano, Beltrán and Shub considered the set of points given by the stereographic projection of the roots of the random elliptic polynomial of degree $n$ and computed the expected logarithmic energy. We study the fluctuations of the logarithmic energy associated to this random configuration and prove a central limit theorem. Our approach shows that all cumulants of the logarithmic energy are asymptotically linear in $n$, and hence the energy is well-concentrated on the scale of $\sqrt{n}$.
In gossip networks, a source node forwards time-stamped updates to a network of observers according to a Poisson process. The observers then update each other on this information according to Poisson processes as well. The Age of Information (AoI) of a given node is the difference between the current time and the most recent time-stamp of source information that the node has received. We provide a method for evaluating the AoI of a node in terms of first passage percolation. We then use this distributional identity to prove matching upper and lower bounds on the AoI in terms of connectivity properties of the underlying network. In particular, if one sets $X_v$ to be the AoI of node $v$ on a finite graph $G$ with $n$ nodes, then we define $m_\ast = \min\{m : m \cdot |B_m(v)| \geq n\}$ where $B_m(v)$ is the ball of radius $m$ in $G$. In the case when the maximum degree of $G$ is bounded by $Δ$ we prove $\mathbb{E} X_v = Θ_Δ(m_\ast)$. As corollaries, we solve multiple open problems in the literature such as showing the age of information on a subset of $\mathbb{Z}^d$ is $Θ(n^{1/(d+1)})$. We also demonstrate examples of graphs with AoI scaling like $n^α$ for each $α\in (0,1/2)$. These graphs are not vertex-transitive and in fact we show that if one considers the AoI on a graph coming from a vertex-transitive infinite graph then either $\mathbb{E} X_v = Θ(n^{1/k})$ for some integer $k \geq 2$ or $\mathbb{E} X_v = n^{o(1)}$.
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.