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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We solve an asymptotic variant of a smooth version of the $A_n$-realization problem for plane curves. As an application, we determine the cobordism distance between torus links of type $T(d,d)$ and $T(2,N)$ up to an error of at most $3d$. We also discuss the limits of knot theoretic approaches aimed at solving the $A_n$-realization problem.

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We introduce and analyse the almost sure convergence of a new stochastic algorithm for the global minimization of Morse functions on compact Riemannian manifolds. This diffusion process is called fraudulent because it requires the knowledge of minimal value of the function to minimize. Its investigation is nevertheless important, since in particular it appears as the limit behavior of non-fraudulent and time-inhomogeneous swarm mean-field algorithms used in global optimization.

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For every integer $g \ge 2$ we show the existence of a compact Riemann surface $\Sigma $ of genus $g$ such that the rank two trivial holomorphic vector bundle $\mathcal{O}^{\oplus 2}_{\Sigma }$ admits holomorphic connections with $\operatorname{SL}(2,\mathbb{R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. This also answers a question of Calsamiglia, Deroin, Heu and Loray. The construction carries over to all very stable and compatible real holomorphic structures over the topologically trivial rank two bundle on $\Sigma $, and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

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The infinite-bin model is a one-dimensional particle system on $\mathbb{Z}$ introduced by Foss and Konstantopoulos in relation with last passage percolation on complete directed acyclic graphs. In this model, at each integer time, a particle is selected at random according to its rank and produces a child at the location immediately to its right. In this article, we consider the limiting distribution of particles after an infinite number of branching events have occurred. Under mild assumptions, we prove that the event (called freezing) that a location contains only a finite number of balls satisfies a $0-1$ law and we provide various criteria to determine whether freezing occurs.

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We consider the quasi-classical limit of Nelson-type regularized polaron models describing a particle interacting with a quantized bosonic field. We break translation-invariance by adding an attractive external potential decaying at infinity, acting on the particle. In the strong coupling limit where the field behaves classically we prove that the model’s energy quasi-minimizers strongly converge to ground states of the limiting Pekar-like non-linear model. This holds for arbitrarily small external attractive potentials, hence this binding is fully due to the interaction with the bosonic field. We use a new approach to the construction of quasi-classical measures to revisit energy convergence, and a localization method in a concentration-compactness type argument to obtain convergence of states.

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We attack the question of $E_2$-formality of differential graded algebras over $\mathbb{F}_p$ via obstruction theory. We are able to prove that $E_2$-algebras whose cohomology ring is a polynomial algebra on even degree classes are formal. As a consequence we prove $E_2$-formality of the classifying space of some compact Lie groups or of Davis–Januszkiewicz spaces.

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In the search of a projective analog of Kunz’s theorem and a Frobenius-theoretic analog of Mori–Hartshorne’s theorem, we investigate the positivity of the kernel of the Frobenius trace (equivalently, the negativity of the cokernel of the Frobenius endomorphism) on a smooth projective variety over an algebraically closed field of positive characteristic. For instance, such a kernel is ample for projective spaces. Conversely, we show that for curves, surfaces, and threefolds the Frobenius trace kernel is ample only for Fano varieties of Picard rank $1$.

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This article constructs Von Neumann invariants for constructible complexes and coherent $\mathcal{D}$-modules on compact complex manifolds, generalizing the work of the author on coherent $L_2$-cohomology. We formulate a conjectural generalization of Dingoyan’s $L_2$-Mixed Hodge structures in terms of Saito’s Mixed Hodge Modules and give partial results in this direction.

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Using the global Kuranishi charts constructed by Hirschi–Swaminathan, we define gravitational descendants and equivariant Gromov–Witten invariants for general symplectic manifolds. We prove that these invariants satisfy the axioms of Kontsevich and Manin and their generalisations. A virtual localisation formula holds in this setting; we use it to derive an explicit formula for the equivariant Gromov–Witten invariants of Hamiltonian GKM manifolds. In particular, the symplectic Gromov–Witten invariants of smooth toric varieties agree with their algebro-geometric counterpart. In the semipositive case, the invariants studied here recover those of Ruan and Tian.

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We study the inviscid Burgers equation on the circle $\mathbb{T}:= \mathbb{R}/\mathbb{Z}$ forced by the spatial derivative of a Poisson point process on $ \mathbb{R}\times \mathbb{T}$. We construct global solutions with mean $\theta $ simultaneously for all $\theta \in \mathbb{R}$, and in addition construct their associated global shocks (which are unique except on a countable set of $\theta $). We then show that as $\theta $ changes, the solution only changes through the movement of the global shock, and give precise formulas for this movement. This can be seen as an analogue of previous results by the author and Yu Gu in the viscous case with white-in-time forcing, which related the derivative of the solution in $\theta $ to the density of a particle diffusing in the Burgers flow.

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Caffarelli’s contraction theorem bounds the derivative of the optimal transport map between a log-convex measure and a strongly log-concave measure. We show that an analogous phenomenon holds on the level of the trace: the trace of the derivative of the optimal transport map between a log-subharmonic measure and a strongly log-concave measure is bounded. We show that this trace bound has a number of consequences pertaining to volume-contracting transport maps, majorization and its monotonicity along Wasserstein geodesics, growth estimates of log-subharmonic functions, the Wehrl conjecture for Glauber states, and two-dimensional Coulomb gases. We also discuss volume-contraction properties for the Kim–Milman transport map.

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We study quasi-isometric representations of finitely generated non-abelian free groups into some higher rank semi-simple Lie groups which are not Anosov, nor approximated by Anosov. We show in some cases that these can be perturbed to be non-quasi-isometric, or to have some instability properties with respect to their action on the flag space.

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The systole of a contact form $\alpha $ is defined as the shortest period of closed Reeb orbits of $\alpha $. Given a non-trivial $\mathbb{S}^1$-principal bundle over $\mathbb{S}^2$ with total space $M$, we prove a sharp systolic inequality for the class of tight contact forms on $M$ invariant under the $\mathbb{S}^1$-action. This inequality exhibits a behavior which depends on the Euler class of the bundle in a subtle way. As applications, we prove a sharp systolic inequality for rotationally symmetric Finsler metrics on $\mathbb{S}^2$, a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.

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In a recent paper, Aldous, Blanc and Curien asked which distributions can be expressed as the distance between two independent random variables on some separable measured metric space. We show that every nonnegative discrete distribution whose support contains $0$ arises in this way, as well as a class of compactly supported distributions with density.

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We introduce the notion of a flat extension of a connection $\theta $ on a principal bundle. Roughly speaking, $\theta $ admits a flat extension if it arises as the pull-back of a component of a Maurer–Cartan form. For trivial bundles over closed oriented $3$-manifolds, we relate the existence of certain flat extensions to the vanishing of the Chern–Simons invariant associated with $\theta $. As an application, we recover the obstruction of Chern–Simons for the existence of a conformal immersion of a Riemannian $3$-manifold into Euclidean $4$-space. In addition, we obtain corresponding statements for a Lorentzian $3$-manifold, as well as a global obstruction for the existence of an equiaffine immersion into $\mathbb{R}^4$ of a $3$-manifold that is equipped with a torsion-free connection preserving a volume form.

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Given a probability measure $\mu $ on a finitely generated group $\Gamma $, the Green function $G(x,y|r)$ encodes many properties of the random walk associated with $\mu $. Endowing $\Gamma $ with a word distance, we denote by $H_r(n)$ the sum of the Green function $G(e,x|r)$ along the sphere of radius $n$. This quantity appears naturally when studying asymptotic properties of branching random walks driven by $\mu $ on $\Gamma $. Our main result exhibits a relatively hyperbolic group with convergent Poincaré series generated by $H_r(n)$.

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We introduce a multi-scale generalization of the notion of scalar product on real and complex vector spaces and study the corresponding Voronoi tilings. This framework allows to analyze limiting metric structures involving several levels of convergence. In particular, we describe metric degenerations of polarized tori and Hausdorff limits of Voronoi tilings.

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