Dolgopyat and Sarig showed that for any piecewise smooth function $f:𝕋\to ℝ$ and almost every pair $(\alpha ,x_0)\in 𝕋\times 𝕋$, $S_N(f,\alpha ,x_0):={\sum }_{n=1}^{N}f(n\alpha +x_0)$ fails to fulfill a temporal distributional limit theorem. In this article, we show that the doubly metric statement can be sharpened to a single metric one: For almost every $\alpha \in 𝕋$ and all $x_0\in 𝕋$, $S_N(f,\alpha ,x_0)$ does not satisfy a temporal distributional limit theorem, regardless of centering and scaling. The obtained results additionally lead to progress in a question posed by Dolgopyat and Sarig.

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The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their distance to equilibrium remains close to the maximal value for a while and suddenly drops to zero as the time parameter reaches a critical threshold. Despite the accumulation of many examples, this phenomenon is still far from being understood, and identifying the general conditions that trigger it has become one of the biggest challenges in the quantitative analysis of finite Markov chains. Very recently, the author proposed a general sufficient condition for the occurrence of a cutoff, based on a certain information-theoretical statistics called varentropy. In the present paper, we demonstrate the sharpness of this approach by showing that the cutoff phenomenon is actually equivalent to the varentropy criterion for all sparse, fast-mixing chains. Reversibility is not required.

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We prove that the closed orbit of the Eierlegende Wollmilchsau is the only ${\text{SL}}_2\left(ℝ\right)$-orbit closure in genus three with a zero Lyapunov exponent in its Kontsevich–Zorich spectrum. The result recovers previous partial results in this direction by Bainbridge–Habegger–Möller and the first named author. The main new contribution is the identification of the differentials in the Hodge bundle corresponding to the Forni subspace in terms of the degenerations of the surface. We use this description of the differentials in the Forni subspace to evaluate them on absolute homology curves and apply the jump problem from the work of Hu and the third named author to the differentials near the boundary of the orbit closure. This results in a simple geometric criterion that excludes the existence of a Forni subspace.

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We show the existence of a trace process at infinity for random walks on hyperbolic groups of conformal dimension $<2$ and relate it to the existence of a reflected random walk. To do so, we employ the theory of Dirichlet forms which connects the theory of symmetric Markov processes to functional analytic perspectives. We introduce a family of Besov spaces associated to random walks and prove that they are isomorphic to some of the Besov spaces constructed from the co-homology of the group studied in Bourdon–Pajot (2003). We also study the regularity of harmonic measures of random walks on hyperbolic groups using the potential theory associated to Dirichlet forms.

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We study the sizes of the Voronoi cells of $k$ uniformly chosen vertices in a random split tree of size $n$. We prove that, for $n$ large, the largest of these $k$ Voronoi cells contains most of the vertices, while the sizes of the remaining ones are essentially all of order $nexp(-\mathrm{const}\sqrt{logn})$. This “winner-takes-all” phenomenon persists if we modify the definition of the Voronoi cells by (a) introducing random edge lengths (with suitable moment assumptions), and (b) assigning different “influence” parameters (called “speeds” in the paper) to each of the $k$ vertices. Our findings are in contrast to corresponding results on random uniform trees and on the continuum random tree, where it is known that the vector of the relative sizes of the $k$ Voronoi cells is asymptotically uniformly distributed on the $(k-1)$-dimensional simplex.

Two intermediary steps in the proof of our main result may be of independent interest because of the information they give on the typical shape of large random split trees: we prove convergence in probability of their “profile”, and we prove asymptotic results for the size of fringe trees (trees rooted at an ancestor of a uniform random node).

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We consider a magnetic Laplacian on a compact manifold, with a constant non-degenerate magnetic field. In the large field limit, it is known that the eigenvalues are grouped in clusters, the corresponding sums of eigenspaces being called the Landau levels. The first level has been studied in-depth as a natural generalization of the Kähler quantization. The current paper is devoted to the higher levels: we compute their dimensions as Riemann–Roch numbers, study the associated Toeplitz algebras and prove that each level is isomorphic with a quantization twisted by a convenient auxiliary bundle.

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We introduce the concept of directed orbifold, namely triples $(X,V,D)$ formed by a directed algebraic or analytic variety $(X,V)$, and a ramification divisor $D$, where $V$ is a coherent subsheaf of the tangent bundle $T_X$. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by $D$ and multiplicities are bounded by the ramification indices of the components of $D$. We estimate precisely the curvature tensor of the corresponding directed structure $V\langle D\rangle$ in the general orbifold case – with a special attention to the compact case $D=0$ and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green–Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.

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We consider the null-controllability problem for the generalized Baouendi–Grushin equation $({\partial }_t-{\partial }_x^2-q{\left(x\right)}^2{\partial }_y^2)f={\mathbb{1}}_{\omega }u$ on a rectangular domain. Sharp controllability results already exist when the control domain $\omega$ is a vertical strip, or when $q\left(x\right)=x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $\omega$. In some geometries for $\omega$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability.

Our proof relies on several tools: known results when $\omega$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-{\partial }_x^2+{\nu }^{2}q{\left(x\right)}^2$ when $Re\left(\nu \right)>0$, pseudo-differential-type operators on polynomials and Runge’s theorem for the lower bound.

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We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending the investigation of the quadratic case in one or more potentials by Leplaideur and Watbled.

We prove a variational principle for the nonlinear pressure and we characterize the nonlinear equilibrium measures and relate them to specific classical equilibrium measures.

In this non-linear thermodynamical formalism, as for mean-field theories of statistical mechanics, several kind of phase transitions appear, some of which cannot happen in the linear case. Our techniques can deal with known cases (Curie–Weiss and Potts models) as well as with new examples (metastable phase transition).

Finally, we apply some of these ideas to the classical, linear setting proving that freezing phase transitions can occur over any zero-entropy invariant compact subset of the phase space.

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The Crump–Young model consists of two fully coupled stochastic processes modeling the substrate and micro-organisms dynamics in a chemostat. Substrate evolves following an ordinary differential equation whose coefficients depend of micro-organisms number. Micro-organisms are modeled though a pure jump process whose jump rates depend on the substrate concentration.

It goes to extinction almost-surely in the sense that micro-organism population vanishes. In this work, we show that, conditionally on the non-extinction, its distribution converges exponentially fast to a quasi-stationary distribution.

Due to the deterministic part, the dynamics of the Crump–Young model are highly degenerated. The proof is therefore original and consists of technically precise estimates and new approaches for quasi-stationary convergence.

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We prove some sharp regularity results for solutions of classical first order hyperbolic initial boundary value problems. Our two main improvements on the existing literature are weaker regularity assumptions for the boundary data and regularity in fractional Sobolev spaces. This last point is specially interesting when the regularity index belongs to $1/2+\mathbb{N}$, as it involves nonlocal compatibility conditions.

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We show that in large enough rank, the Gromov boundary of the free factor complex is path connected and locally path connected.

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We prove that any unimodular Pisot substitution subshift is measurably conjugate to a domain exchange in a Euclidean space which is a finite topological extension of a translation on a torus. This generalizes the pioneer works of Rauzy and Arnoux–Ito providing geometric realizations to any unimodular Pisot substitution without any additional combinatorial condition.

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In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers of torus knot complements. We determine their Benjamini–Schramm limit and the linear growth rate of the Betti numbers of these covers. All these results generalise to a larger class of lattices in $\mathrm{PLS}(2,ℝ)\times ℝ$. As a by-product of our proofs, we obtain analogous limit theorems for high index random subgroups of non-uniform Fuchsian lattices with torsion.

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We study the asymptotic behaviour of a version of the one-dimensional Mott random walk in a regime that exhibits severe blocking. We establish that, for any fixed time, the appropriately-rescaled Mott random walk is situated between two environment-measurable barriers, the locations of which are shown to have an extremal scaling limit. Moreover, we give an asymptotic description of the distribution of the Mott random walk between the barriers that contain it.

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