Permalinkhttps://doi.org/10.5802/ahl.135

Keywordsstochastic individual-based model, spine of birth-death process, many-to-one formula, ancestral path, historical process, genealogy, phylogeny, resilience

Abstract

We investigate a simple quantitative genetics model subject to a gradual environmental change from the viewpoint of the phylogenies of the living individuals. We aim to understand better how the past traits of their ancestors are shaped by the adaptation to the varying environment. The individuals are characterized by a one-dimensional trait. The dynamics -births and deaths- depend on a time-changing mortality rate that shifts the optimal trait to the right at constant speed. The population size is regulated by a nonlinear non-local logistic competition term. The macroscopic behaviour can be described by a PDE that admits a unique positive stationary solution. In the stationary regime, the population can persist, but with a lag in the trait distribution due to the environmental change. For the microscopic (individual-based) stochastic process, the evolution of the lineages can be traced back using the historical process, that is, a measure-valued process on the set of continuous real functions of time. Assuming stationarity of the trait distribution, we describe the limiting distribution, in large populations, of the path of an individual drawn at random at a given time $T$. Freezing the non-linearity due to competition allows the use of a many-to-one identity together with Feynman–Kac’s formula. This path, in reversed time, remains close to a simple Ornstein–Uhlenbeck process. It shows how the lagged bulk of the present population stems from ancestors once optimal in trait but still in the tail of the trait distribution in which they lived.

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Permalinkhttps://doi.org/10.5802/ahl.134

KeywordsVlasov–Poisson system, reflection boundary conditions, generalized characteristics

Abstract

The Vlasov–Poisson system models a collisionless plasma. In a bounded domain it is known that singularities can occur. Existence of global in time continuous solutions to the Vlasov–Poisson system is proven in a one-dimensional bounded domain, with direct reflection boundary conditions and initial data even with respect to the $v$-variable. Local in time uniqueness is proven. Generalized characteristics are used. Electroneutrality is obtained in the limit.

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Permalinkhttps://doi.org/10.5802/ahl.133

KeywordsAbelian variety, Frobenius eigenvalue, distribution, isogeny, complex multiplication, Katz–Sarnak

Abstract

We show that for a large class of rings $R$, the number of principally polarized abelian varieties over a finite field in a given simple ordinary isogeny class and with endomorphism ring $R$ is equal either to $0$, or to a ratio of class numbers associated to $R$, up to some small computable factors. This class of rings includes the maximal order of the CM field $K$ associated to the isogeny class (for which the result was already known), as well as the order $R$ generated over $\mathbf{Z}$ by Frobenius and Verschiebung.

For this latter order, we can use results of Louboutin to estimate the appropriate ratio of class numbers in terms of the size of the base field and the Frobenius angles of the isogeny class. The error terms in our estimates are quite large, but the trigonometric terms in the estimate are suggestive: Combined with a result of Vlăduţ on the distribution of Frobenius angles of isogeny classes, they give a heuristic argument in support of the theorem of Katz and Sarnak on the limiting distribution of the multiset of Frobenius angles for principally polarized abelian varieties of a fixed dimension over finite fields.

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Permalinkhttps://doi.org/10.5802/ahl.132

KeywordsIrreducible lattice, groups acting on trees, just-infinite group, $2$-transitive group

Abstract

Let $T_1,T_2$ be regular trees of degrees $d_1,d_2\ge 3$. Let also $\Gamma \le \mathrm{Aut}\left(T_1\right)\times \mathrm{Aut}\left(T_2\right)$ be a group acting freely and transitively on $V{T}_1\times V{T}_2$. For $i=1$ and $2$, assume that the local action of $\Gamma$ on $T_i$ is $2$-transitive; if moreover $d_i\ge 7$, assume that the local action contains $\mathrm{Alt}\left(d_i\right)$. We show that $\Gamma$ is irreducible, unless $(d_1,d_2)$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $\Gamma$ that can be checked purely in terms of its local action on a ball of radius $1$ in $T_1$ and $T_2$. Under the same hypotheses, we show moreover that if $\Gamma$ is irreducible, then it is hereditarily just-infinite, provided the local action on $T_i$ is not the affine group ${\mathbf{F}}_5⋊{\mathbf{F}}_5^{*}$. The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.

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Permalinkhttps://doi.org/10.5802/ahl.131

KeywordsHeavy atoms, ground state density, relativistic Coulomb system, Scott conjecture

Abstract

We prove the convergence of the density on the scale $Z^{-1}$ to the density of the Bohr atom with infinitely many electrons (strong Scott conjecture) for a model that is known to describe heavy atoms accurately.

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Permalinkhttps://doi.org/10.5802/ahl.130

KeywordsRandom walk; cone; exit time; Green function; harmonic function; Martin boundary; Brownian motion; coupling

Abstract

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

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Permalinkhttps://doi.org/10.5802/ahl.129

KeywordsHypocoercivity, Langevin dynamics, Schur complements, resolvent estimates

Abstract

We propose an approach to obtaining explicit estimates on the resolvent of hypocoercive operators by using Schur complements, rather than from an exponential decay of the evolution semigroup combined with a time integral. We present applications to Langevin-like dynamics and Fokker–Planck equations, as well as the linear Boltzmann equation (which is also the generator of randomized Hybrid Monte Carlo in molecular dynamics). In particular, we make precise the dependence of the resolvent bounds on the parameters of the dynamics and on the dimension. We also highlight the relationship of our method with other hypocoercive approaches.

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Permalinkhttps://doi.org/10.5802/ahl.128

KeywordsHighly transitive actions, infinite permutation groups, confined subgroups and uniformly recurrent subgroups, groups of piecewise linear homeomorphisms, Higman–Thompson groups

Abstract

An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k\ge 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgroup of $G$, then the action of $H$ remains highly transitive, possibly after discarding finitely many points.

This result provides a tool to rule out the existence of highly transitive actions, and to classify highly transitive actions of a given group. We give concrete illustrations of these applications in the realm of groups of dynamical origin. In particular we obtain the first non-trivial classification of highly transitive actions of a finitely generated group.

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Permalinkhttps://doi.org/10.5802/ahl.127

KeywordsHartree equation, nonlinear Schrödinger equation, density matrices, random fields, stability, scattering

Abstract

We prove a scattering result near certain steady states for a Hartree equation for a random field. This equation describes the evolution of a system of infinitely many particles. It is an analogous formulation of the usual Hartree equation for density matrices. We treat dimensions $2$ and $3$, extending our previous result. We reach a large class of interaction potentials, which includes the nonlinear Schrödinger equation. This result has an incidence in the density matrices framework. The proof relies on dispersive techniques used for the study of scattering for the nonlinear Schrödinger equation, and on the use of explicit low frequency cancellations as done by Lewin and Sabin. To relate to density matrices, we use Strichartz estimates for orthonormal systems from Frank and Sabin, and Leibniz rules for integral operators.

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Permalinkhttps://doi.org/10.5802/ahl.126

Keywordsnormalized gradient flow, ground states, stationary states, quantum graphs, nonlinear Schrödinger equation

Abstract

We introduce and implement a method to compute stationary states of nonlinear Schrödinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schrödinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.

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Permalinkhttps://doi.org/10.5802/ahl.125

KeywordsRandom maps, random trees, scaling limits

Abstract

We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit nontrivial subsequential limits as $n\to \infty$ in the Gromov–Hausdorff–Prokhorov topology. Further, we show that they converge in distribution towards the celebrated Brownian sphere, and more generally a Brownian disk for maps with a boundary, if and only if there is no inner face with a macroscopic degree, or, if the perimeter is too big, the maps degenerate and converge to the Brownian tree. By first sampling the degrees at random with an appropriate distribution, this model recovers that of size-conditioned Boltzmann maps associated with critical weights in the domain of attraction of a stable law with index $\alpha \in [1,2]$. The Brownian tree and disks then appear respectively in the case $\alpha =1$ and $\alpha =2$, whereas in the case $\alpha \in (1,2)$ our results partially recover previous known ones. Our proofs rely on known bijections with labelled plane trees, which are similarly sampled uniformly at random given $n$ outdegrees. Along the way, we obtain some results on the geometry of such trees, such as a convergence to the Brownian tree but only in the weaker sense of subtrees spanned by random vertices, which are of independent interest.

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Permalinkhttps://doi.org/10.5802/ahl.124

KeywordsSchreier decoration, Borel graphs, local-global equivalence, edge colorings

Abstract

Following recent result of L. M. Tóth [Tót21, Annales Henri Lebesgue, Volume 4 (2021)] we show that every $2\Delta$-regular Borel graph $𝒢$ with a (not necessarily invariant) Borel probability measure admits approximate Schreier decoration. In fact, we show that both ingredients from the analogous statements for finite graphs have approximate counterparts in the measurable setting, i.e., approximate Kőnig’s line coloring Theorem for Borel graphs without odd cycles and approximate balanced orientation for even degree Borel graphs.

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Permalinkhttps://doi.org/10.5802/ahl.123

Keywordsgrowth-fragmentation equation, self-similar fragmentation, measure solutions, long-time behavior, general relative entropy, Harris’s theorem, periodic semigroups

Abstract

We are interested in a non-local partial differential equation modeling equal mitosis. We prove that the solutions present persistent asymptotic oscillations and that the convergence to this periodic behavior, in suitable spaces of weighted signed measures, occurs exponentially fast. It can be seen as a spectral gap result between the countable set of dominant eigenvalues and the rest of the spectrum, which is to our knowledge completely new. The two main difficulties in the proof are to define the projection onto the subspace of periodic (rescaled) solutions and to estimate the speed of convergence to this projection. The first one is addressed by using the generalized relative entropy structure of the dual equation, and the second is tackled by applying Harris’s ergodic theorem on sub-problems.

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Permalinkhttps://doi.org/10.5802/ahl.122

KeywordsStochastic regularization, Stochastic Partial Differential Equations, Nonlinear Schrödinger equation, Hardy–Littlewood–Sobolev inequality

Abstract

In this paper, we investigate a stochastic Hardy–Littlewood–Sobolev inequality. Due to the non-homogenous nature of the potential in the inequality, we show that a constant proportional to the length of the interval appears on the right-hand-side. As a direct application, we derive local Strichartz estimates for randomly modulated dispersions and solve the Cauchy problem of the critical nonlinear Schrödinger equation.

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Permalinkhttps://doi.org/10.5802/ahl.121

KeywordsDouble bracket, Hamiltonian algebra, Quasi-Hamiltonian algebra, Non-commutative geometry, Integrable system

Abstract

Double (quasi-)Poisson algebras were introduced by Van den Bergh as non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket. In this work, we provide a study of morphisms of double (quasi-)Poisson algebras, which we relate to the $H_0$-Poisson structures of Crawley–Boevey. We prove in particular that the double (quasi-) Poisson algebra structure defined by Van den Bergh for an arbitrary quiver only depends upon the quiver seen as an undirected graph, up to isomorphism. We derive from our results a representation theoretic description of action-angle duality for several classical integrable systems.

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