arXiv subjects

Luke Peilen

Publications and source records attributed to Luke Peilen.

9 recordsLinked to original sources

arXiv2025

Local Laws and Fluctuations for Super-Coulombic Riesz Gases

We study the local statistical behavior of the super-Coulombic Riesz gas of particles in Euclidean space of arbitrary dimension, with inverse power distance repulsion integrable near $0$, and with a general confinement potential, in a certain regime of inverse temperature. Using a bootstrap procedure, we prove local laws on the next order energy and control on fluctuations of linear statistics that are valid down to the microscopic lengthscale, and provide controls for instance, on the number of particles in a (mesoscopic or microscopic) box, and the existence of a limit point process up to subsequences. As a consequence of the local laws, we derive an almost additivity of the free energy that allows us to exhibit for the first time a CLT for Riesz gases corresponding to small enough inverse powers, at small mesoscopic length scales, which can be interpreted as the convergence of the associated potential to a fractional Gaussian field. Compared to the Coulomb interaction case, the main new issues arise from the nonlocal aspect of the Riesz kernel. This manifests in (i) a novel technical difficulty in generalizing the transport approach of Lebl\'e and the second author to the Riesz gas which now requires analyzing a degenerate and singular elliptic PDE, (ii) the fact that the transport map is not localized, which makes it more delicate to localize the estimates, (iii) the need for coupling the local laws and the fluctuations control inside the same bootstrap procedure.

math-ph↗

arXiv2025

Configurations of 10 points and their incidence varieties

Incidence varieties are spaces of $n$-tuples of points in the projective plane that satisfy a given set of collinearity conditions. We classify the components of incidence varieties and realization moduli spaces associated to configurations of up to 10 points, up to birational equivalence. We show that each realization space component is birational to a projective space, a genus 1 curve, or a K3 surface. To do this, we reduce the problem to a study of 163 special arrangements called superfigurations. Then we use computer algebra to describe the realization space of each superfiguration.

math.AG↗

arXiv2025

Poisson Statistics for Coulomb Gases at Intermediate Temperature Regimes

We consider the microscopic statistics of a Coulomb gas in $\mathbb{R}^2$ at intermediate temperatures. In particular, we show that the microscopic point process associated to the Coulomb gas converges to a homogeneous Poisson point process at intermediate temperature regimes $\beta N \rightarrow \infty$ and $\beta \sqrt{N} \log N \rightarrow 0$, extending previous results. Our approach relies on a novel quantitative asymptotic description of correlation functions, which is of its own interest.

math.PR↗

arXiv2024

Emergence of a Poisson process in weakly interacting particle systems

We consider the Gibbs measure of a general interacting particle system for a certain class of ``weakly interacting" kernels. In particular, we show that the local point process converges to a Poisson point process as long as the inverse temperature $\beta$ satisfies $N^{-1} \ll \beta \ll N^{-\frac{1}{2}}$, where $N$ is the number of particles. This expands the temperature regime for which convergence to a Poisson point process has been proved.

math.PR↗

arXiv2024

On the Maximum of the Potential of a General Two-Dimensional Coulomb Gas

We determine the leading order of the maximum of the random potential associated to a two-dimensional Coulomb gas for general $\beta$ and general confinement potential, extending the recent result of Lambert-Lebl\'e-Zeitouni. In the case $\beta=2$, this corresponds to the (centered) log-characteristic polynomial of either the Ginibre random matrix ensemble for $V(x)=\frac{|x|^2}{2}$ or a more general normal matrix ensemble. The result on the leading order asymptotics for the maximum of the log-characteristic polynomial is new for random normal matrices. We rely on connections with the classical obstacle problem and the theory of Gaussian Multiplicative Chaos. We make use of a new concentration result for fluctuations of $C^{1,1}$ linear statistics which may be of independent interest.

math.PR↗

arXiv2022

Local Laws and a Mesoscopic CLT for $β$-ensembles

We study the statistical mechanics of the log-gas, or $β$-ensemble, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on the next order energy that are valid down to microscopic length scales. To our knowledge, this is the first time that this kind of a local quantity has been controlled for the log-gas. Simultaneously, we exhibit a control on fluctuations of linear statistics that is valid at all mesoscales. Using these local laws, we are able to exhibit for the first time a CLT at arbitrary mesoscales, improving upon a previous result of Bekerman-Lodhia that was true only for power mesoscales.

math-ph↗

arXiv2019

Topology of tropical moduli of weighted stable curves

The moduli space $Δ_{g,w}$ of tropical $w$-weighted stable curves of volume $1$ is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of $w$-weighted stable curves. If at least two of the weights are $1$, we prove that $Δ_{0,w}$ is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space $Δ_{0,w}$ is disconnected and for which the fundamental group of $Δ_{0,w}$ has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri-Hampe-Markwig-Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus $0$ and $1$, and leverage this to extend several of our genus $0$ results to the spaces $Δ_{1,w}$.

math.CO↗

arXiv2018

The cycle structure of a Markoff automorphism over finite fields

We begin an investigation of the action of pseudo-Anosov elements of $\mathrm{Out}(\mathbf{F}_{2})$ on the Markoff-type varieties \[ \mathbb{X}_κ:\:x^{2}+y^{2}+z^{2}=xyz+2+κ\] over finite fields $\mathbb{F}_{p}$ with $p$ prime. We first make a precise conjecture about the permutation group generated by $\mathrm{Out}(\mathbf{F}_{2})$ on $\mathbb{X}_{-2}(\mathbb{F}_{p})$ that shows there is no obstruction at the level of the permutation group to a pseudo-Anosov acting `generically'. We prove that this conjecture is sharp. We show that for a fixed pseudo-Anosov $g\in\mathrm{Out}(\mathbf{F}_{2})$, there is always an orbit of $g$ of length $\geq C\log p+O(1)$ on $\mathbb{X}_κ(\mathbb{F}_{p})$ where $C>0$ is given in terms of the eigenvalues of $g$ viewed as an element of $\mathrm{GL}_{2}(\mathbf{Z})$. This improves on a result of Silverman (2007) that applies to general morphisms of quasi-projective varieties. We have discovered that the asymptotic $(p\to\infty)$ behavior of the longest orbit of a fixed pseudo-Anosov $g$ acting on $\mathbb{X}_{-2}(\mathbb{F}_{p})$ is dictated by a dichotomy that we describe both in combinatorial terms and in algebraic terms related to Gauss's ambiguous binary quadratic forms, following Sarnak. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture.

math.NT↗

arXiv2017

Counting Arcs in Projective Planes via Glynn's Algorithm

An $n$-arc in a projective plane is a collection of $n$ distinct points in the plane, no three of which lie on a line. Formulas counting the number of $n$-arcs in any finite projective plane of order $q$ are known for $n \le 8$. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted $9$-arcs in the projective plane over a finite field of order $q$ and showed that this count is a quasipolynomial function of $q$. We present a formula for the number of $9$-arcs in any projective plane of order $q$, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as a special case. We obtain our formula from a new implementation of an algorithm due to Glynn; we give details of our implementation and discuss its consequences for larger arcs.

math.CO↗