Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group $\Gamma $. In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of $\Gamma $ according to one of the metrics. Our results also apply to various other metrics including length functions associated to Anosov representations and to group actions on hyperbolic metric spaces.

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In 2000, Margulis proved that any group of homeomorphisms of the circle either preserves a probability measure on the circle or contains a free subgroup on two generators, which is reminiscent of the Tits alternative for linear groups. In this article, we prove an analogous statement for groups of locally monotonic homeomorphisms of a compact subset of $\mathbb{R}$. The proof relies on dynamical properties of random walks on the group, which may be of independent interest.

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We study the distribution of the angles between Oseledets subspaces and their log-integrability, focusing on dimension $2$. For random i.i.d. products of matrices, we construct examples of probability measures on $\operatorname{GL}_2(\mathbb{R})$ with finite first moment where the Oseledets angle is not log-integrable. We also show that for probability measures with finite second moment the angle is always log-integrable. We then consider general measurable $\operatorname{GL}_2(\mathbb{R})$-cocycles over an arbitrary ergodic automorphism of a non-atomic Lebesgue space, proving that no integrability condition on the matrix distribution ensures log-integrability of the angle. In fact, the joint distribution of the Oseledets spaces can be chosen arbitrarily. A similar flexibility result for bounded cocycles holds under an unavoidable technical restriction.

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Many results of smooth hypoellipticity are available for scalar equations. Much remains to be done for systems and/or at different levels of regularity and in particular for $L^1$-hypoellipticity. In this article we provide some examples and counter-examples.

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We show that there is a rank $1$ transformation that is mildly mixing but does not have minimal self-joinings, answering a question of Thouvenot.

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We are interested in the scattering problem for the cubic 3D nonlinear defocusing Schrödinger equation with variable coefficients. Previous scattering results for such problems address only the cases with constant coefficients or assume strong variants of the non-trapping condition, stating that all the trajectories of the Hamiltonian flow associated with the operator are escaping to infinity. In contrast, we consider the most general setting, where strong trapping, such as stable closed geodesics, may occur, but we introduce a compactly supported damping term localized in the trapping region, to explore how damping can mitigate the effects of trapping.

In addition to the challenges posed by the trapped trajectories, notably the loss of smoothing and of scale-invariant Strichartz estimates, difficulties arise from the damping itself, particularly since the energy is not, a priori, bounded. For $H^{1+\epsilon }$ initial data (chosen because the local-in-time theory is a priori no better than for 3D unbounded manifolds, where local well-posedness of strong $H^1$ solutions is unavailable) we establish global existence and scattering in $H^{s}$ for any $0 \le s <1$ in positive times, the inability to reach $H^1$ being related to the loss of smoothing due to trapping.

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The now classical convergence in distribution theorem for well normalized sums of stationary martingale increments has been extended to multi-indexed martingale increments. In the present article we make progress in the identification of the limit law.

In dimension one, as soon as the stationary martingale increments form an ergodic process, the limit law is normal, and it is still the case for multi-indexed martingale increments when one of the processes defined by one coordinate of the multidimensional time is ergodic. In the general case, the limit may be non normal.

In the present paper we establish links between the dynamical properties of the $\mathbb{Z}^d$-measure preserving action associated to the stationary random field (like the positivity of the entropy of some factors) and the existence of a non normal limit law. The identification of a natural factor on which the $\mathbb{Z}^d$-action is of product type is a crucial step in this approach.

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We interpret the Hilbert entropy of a convex projective structure on a closed higher-genus surface as the Hausdorff dimension of the non-differentiability points of the limit set in the full flag space $\mathcal{F}(\mathbb{R}^3)$. Generalizations for regularity properties of boundary maps between locally conformal representations are also discussed. An ingredient for the proofs is the concept of hyperplane conicality that we introduce for a $\theta $-Anosov representation into a reductive real-algebraic Lie group $\mathsf {G}$. In contrast with directional conicality, hyperplane-conical points always have full mass for the corresponding Patterson–Sullivan measure.

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We investigate the minimum and maximum number of nodal domains across all time-dependent homogeneous caloric polynomials of degree $d$ in $\mathbb{R}^{n}\times \mathbb{R}$ (space $\times $ time), i.e., polynomial solutions of the heat equation satisfying $\partial _t p\lnot \equiv 0$ and

When $n=1$, it is classically known that the number of nodal domains is precisely $2\lceil d/2\rceil $. When $n=2$, we prove that the minimum number of nodal domains is 2 if $d\lnot \equiv 0\pmod {4}$ and is 3 if $d\equiv 0\pmod {4}$. When $n\ge 3$, we prove that the minimum number of nodal domains is $2$ for all $d$. Finally, we show that the maximum number of nodal domains is $\Theta (d^n)$ as $d\rightarrow \infty $ and lies between $\lfloor \frac{d}{n}\rfloor ^n$ and $\binom{n+d}{n}$ for all $n$ and $d$. As an application and motivation for counting nodal domains, we confirm existence of the singular strata in Mourgoglou and Puliatti’s two-phase free boundary regularity theorem for caloric measure.

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In a recent paper [CEG23], Chen, Erchenko and Gogolev have proven that if a Riemannian manifold with boundary has hyperbolic geodesic trapped set, then it can be embedded into a compact manifold whose geodesic flow is Anosov. They have to introduce some assumptions that we discuss here. We explain how some can be removed, obtaining in particular a result applicable to all reasonable 3 dimensional examples.

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We study the configurations of the nearest neighbor Ising ferromagnetic chain with IID centered and square integrable external random field in the limit in which the pairwise interaction tends to infinity. The available free energy estimates for this model show a strong form of disorder relevance (i.e., a strong effect of disorder on the free energy behavior) and our aim is to make explicit how the disorder affects the spin configurations. We give a quantitative estimate that shows that the infinite volume spin configurations are close to one explicit disorder dependent configuration when the interaction is large. Our results confirm predictions on this model obtained by D. S. Fisher and coauthors by applying the renormalization group method.

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At high temperature, the overlap of two particles chosen independently according to the Gibbs measure of the branching Brownian motion converges to zero as time goes to infinity. We investigate the precise decay rate of the probability to obtain an overlap greater than $a$, for some $a>0$, in the whole subcritical phase of inverse temperatures $\beta \in [0,\beta _c)$. Moreover, we study this probability both conditionally on the branching Brownian motion and non-conditionally. Two sub-phases of inverse temperatures appear, but surprisingly the threshold is not the same in both cases.

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We consider large uniform random trees where we fix for each vertex its degree and height. We prove, under natural conditions of convergence for the profile, that those trees properly renormalized converge. To this end, we study the paths from random vertices to the root using coalescent processes. As an application, we obtain scaling limits of Bienaymé–Galton–Watson trees in varying environment.

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We study approaches for compressing the empirical measure in the context of finite dimensional reproducing kernel Hilbert spaces (RKHSs). In this context, the empirical measure is contained within a natural convex set and can be approximated using convex optimization methods. Such an approximation gives rise to a coreset of data points. A key quantity that controls how large such a coreset has to be is the size of the largest ball around the empirical measure that is contained within the empirical convex set. The bulk of our work is concerned with deriving high probability lower bounds on the size of such a ball under various conditions and in various settings: we show how conditions on the density of the data and the kernel function can be used to infer such lower bounds; we further develop an approach that uses a lower bound on the smallest eigenvalue of a covariance operator to provide lower bounds on the size of such a ball; we extend the approach to approximate covariance operators and we show how it can be used in the context of kernel ridge regression. We also derive compression guarantees when standard algorithms like the conditional gradient method are used and we discuss variations of such algorithms to improve the runtime of these standard algorithms. We conclude with a construction of an infinite dimensional RKHS for which the compression is poor, highlighting some of the difficulties one faces when trying to move to infinite dimensional RKHSs.

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We build a general theory of microlocal (homogeneous) Fourier integral operators in real-analytic regularity, following the general construction in the smooth case by Hörmander and Duistermaat. In particular, we prove that the Boutet–Sjöstrand parametrix for the Szegő projector at the boundary of a strongly pseudo-convex real-analytic domain can be realised by an analytic Fourier integral operator. We then study some applications, such as FBI-type transforms on compact, real-analytic Riemannian manifolds and propagators of one-homogeneous (pseudo)differential operators.

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We show that symmetric random walks on non-elementary hyperbolic groups with non-zero homomorphisms into the reals are noise stable at linear scale under finite exponential moment condition.

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