The backward heat flow on the real line started from the initial condition $z^n$ results in the classical $n^{\rm th}$ Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large $n$ limit. Similarly, the backward heat flow with the periodic initial condition $(\sin \frac{\theta }{2})^n$ leads to trigonometric or unitary analogues of the Hermite polynomials. These polynomials are closely related to the partition function of the Curie–Weiss model and appeared in the work of Mirabelli on finite free probability. We relate the $n^{\rm th}$ unitary Hermite polynomial to the expected characteristic polynomial of a unitary random matrix obtained by running a Brownian motion on the unitary group $U(n)$. We identify the global distribution of zeroes of the unitary Hermite polynomials as the free unitary normal distribution. We also compute the asymptotics of these polynomials or, equivalently, the free energy of the Curie–Weiss model in a complex external field. We identify the global distribution of the Lee–Yang zeroes of this model. Finally, we show that the backward heat flow applied to a high-degree real-rooted polynomial (respectively, trigonometric polynomial) induces, on the level of the asymptotic distribution of its roots, a free Brownian motion (respectively, free unitary Brownian motion).

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We consider a recently introduced model of color-avoiding percolation (abbreviated CA-percolation) defined as follows. Every edge in a graph $G$ is colored in some of $k\ge 2$ colors. Two vertices $u$ and $v$ in $G$ are said to be CA-connected if $u$ and $v$ may be connected using any subset of $k-1$ colors. CA-connectivity defines an equivalence relation on the vertex set of $G$ whose classes are called CA-components.

We study the component structure of a randomly colored Erdős–Rényi random graph of constant average degree. We distinguish three regimes for the size of the largest component: a supercritical regime, a so-called intermediate regime, and a subcritical regime, in which the largest CA-component has respectively linear, logarithmic, and bounded size. Interestingly, in the subcritical regime, the bound is deterministic and given by the number of colors.

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Using Givental’s non-linear Maslov index we define a sequence of spectral selectors on the universal cover of the identity component of the contactomorphism group of any lens space. As applications, we prove for lens spaces with equal weights that the standard Reeb flow is a geodesic for the discriminant and oscillation norms, and we define for general lens spaces a stably unbounded conjugation invariant spectral pseudonorm.

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Let us consider subcritical Bernoulli percolation on a connected, transitive, infinite and locally finite graph. In this paper, we propose a new (and short) proof of the exponential decay property for the volume of clusters. We do not rely on differential inequalities and rather use stochastic comparison techniques, which are inspired by several works including the paper An approximate zero-one law written by Russo in the early eighties.

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We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaymé trees. Our bounds are optimal at this level of generality. We also obtain precise height and width asymptotics when the trees’ offspring distributions lie within the domain of attraction of a Cauchy distribution and satisfy a local regularity condition. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaymé trees.

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The reconstruction theorem and the multilevel Schauder estimate have central roles in the analytic theory of regularity structures by Hairer (2014). Inspired by Otto and Weber’s work (2019), we provide elementary proofs for them by using the semigroup of operators. Essentially, we use only the semigroup property and the upper estimates of kernels. Moreover, we refine the several types of Besov reconstruction theorems considered by Hairer–Labbé (2017) and Broux–Lee (2022) and introduce the new framework of “regularity-integrability structures”. The analytic theorems in this paper are applied to the study of quasilinear SPDEs by Bailleul–Hoshino–Kusuoka (2022+) and an inductive proof of the convergence of random models by Bailleul–Hoshino (2023+).

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We consider the Hartree and Schrödinger equations describing the time evolution of wave functions of infinitely many interacting fermions in three-dimensional space. These equations can be formulated using density operators, and they have infinitely many stationary solutions. In this paper, we prove the asymptotic stability of a wide class of stationary solutions. We emphasize that our result includes Fermi gas at zero temperature. This is one of the most important steady states from the physics point of view; however, its asymptotic stability has been left open after the seminal work of Lewin and Sabin [LS14], which first formulated this stability problem and gave significant results.

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Motivated by the study of the convex hull of the trajectory of a Brownian motion in the unit disk reflected orthogonally at its boundary, we study inhomogeneous fragmentation processes in which particles of mass $m \in (0,1)$ split at a rate proportional to $|\log m|^{-1}$. These processes do not belong to the well-studied family of self-similar fragmentation processes. Our main results characterize the Laplace transform of the typical fragment of such a process, at any time, and its large time behavior.

We connect this asymptotic behavior to the prediction obtained by physicists in [BBMS22] for the growth of the perimeter of the convex hull of a Brownian motion in the disc reflected at its boundary. We also describe the large time asymptotic behavior of the whole fragmentation process. In order to implement our results, we make a detailed study of a time-changed subordinator, which may be of independent interest.

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We consider a scattering theory for difference operators on $\mathcal{H}=\ell ^2(\mathbb{Z}^d; \mathbb{C}^n)$ perturbed with a long-range potential $V:\mathbb{Z}^d\rightarrow \mathbb{R}^n$. One of the motivating examples is discrete Schrödinger operators on $\mathbb{Z}^d$-periodic graphs. We construct time-independent modifiers, so-called Isozaki–Kitada modifiers, and we prove that the modified wave operators with the above-mentioned Isozaki–Kitada modifiers exist and that they are complete.

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We begin by presenting a version of the Poincaré–Lefschetz theorem for certain cellular cosheaves on a particular subdivision of a CW-complex $K$. To that end we construct a cellular sheaf on $K$ whose cohomology with compact support is isomorphic to the homology of the initial cosheaf. Thereafter we use the first result to generalise the tropical version of the Lefschetz Hyperplane Section Theorem to some singular tropical toric varieties and singular tropical hypersurfaces.

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Let $S$ be a closed surface of genus $g\ge 2$, furnished with a Borel probability measure $\lambda $ with total support. We show that if $f$ is a $\lambda $-preserving homeomorphism isotopic to the identity such that the rotation vector $\operatorname{rot}_f(\lambda )\in H_1(S,\mathbb{R})$ is a multiple of an element of $H_1(S,\mathbb{Z})$, then $f$ has infinitely many periodic orbits.

Moreover, these periodic orbits can be supposed to have their rotation vectors arbitrarily close to the rotation vector of any fixed ergodic Borel probability measure.

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We completely classify the locally finite, infinite graphs with pure mapping class groups admitting a coarsely bounded generating set. We also study algebraic properties of the pure mapping class group. We establish a semidirect product decomposition, compute first integral cohomology, and classify when they satisfy residual finiteness and the Tits alternative. These results provide a framework and some initial steps towards quasi-isometric and algebraic rigidity of these groups.

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The aim of this article is to derive discontinuous finite elements vector spaces which can be put in a discrete de Rham complex for which the matching between the continuous and discrete cohomology spaces can be proven for periodic meshes.

First, the triangular case is addressed, for which we prove that this property holds for the classical discontinuous finite element space for vectors.

On Cartesian meshes, this result does not hold for the classical discontinuous finite element space for vectors. We then show how to use the de Rham complex found for triangular meshes for enriching the finite element space on Cartesian meshes in order to recover a de Rham complex, on which the same property is proven.

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We study a natural class of Fermi–Ulam Models featuring good hyperbolicity properties that we call dispersing Fermi–Ulam models. Using tools inspired by the theory of hyperbolic billiards we prove, under very mild complexity assumption, a Growth Lemma for our systems. This allows us to obtain ergodicity of dispersing Fermi–Ulam Models. It follows that almost every orbit of such systems is oscillatory.

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