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1. Half-space stationary Kardar–Parisi–Zhang equation beyond the Brownian case

   https://doi.org/10.1088/1751-8121/ac761d  

   Barraquand, Guillaume; Krajenbrink, Alexandre; Le Doussal, Pierre (June 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract We study the Kardar–Parisi–Zhang (KPZ) equation on the half-line x ⩾ 0 with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter u of the dynamics, and the drift − v of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time t , it is natural to rescale parameters as ( u , v ) = t −1/3 ( a , b ) to study the critical region. In the special case a + b = 0, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase ( a < 0) but not in the unbound phase. For instance, starting from the flat or droplet initial condition, the height field near the boundary converges to the stationary process with a > 0 and b = 0, which is not Brownian. For a + b ⩾ 0, we determine exactly the large time distribution F a , b stat of the height function h (0, t ). As an application, we obtain the exact covariance of the height field in a half-line at two times 1 ≪ t 1 ≪ t 2 starting from stationary initial condition, as well as estimates, when starting from droplet initial condition, in the limit t 1 / t 2 → 1. 

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2. Rates of mixing for the measure of maximal entropy of dispersing billiard maps

   https://doi.org/10.1112/plms.12578  

   Demers, Mark F.; Korepanov, Alexey (December 2023, Proceedings of the London Mathematical Society)

   Abstract In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super‐polynomial rate. 

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3. Hölder Parameterization of Iterated Function Systems and a Self-Aflne Phenomenon

   https://doi.org/10.1515/agms-2020-0125  

   Badger, Matthew; Vellis, Vyron (January 2021, Analysis and Geometry in Metric Spaces)

   null (Ed.)

   Abstract We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/ s )-Hölder path-connected, where s is the similarity dimension of the IFS. Then we show that every connected attractor of an IFS is parameterized by a (1/ α)-Hölder curve for all α > s . At the endpoint, α = s , a theorem of Remes from 1998 already established that connected self-similar sets in Euclidean space that satisfy the open set condition are parameterized by (1/ s )-Hölder curves. In a secondary result, we show how to promote Remes’ theorem to self-similar sets in complete metric spaces, but in this setting require the attractor to have positive s -dimensional Hausdorff measure in lieu of the open set condition. To close the paper, we determine sharp Hölder exponents of parameterizations in the class of connected self-affine Bedford-McMullen carpets and build parameterizations of self-affine sponges. An interesting phenomenon emerges in the self-affine setting. While the optimal parameter s for a self-similar curve in ℝ n is always at most the ambient dimension n , the optimal parameter s for a self-affine curve in ℝ n may be strictly greater than n . 

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4. On the Quenched CLT for Stationary Markov Chains

   https://doi.org/10.1007/s10959-023-01241-0  

   Peligrad, Magda (May 2023, Journal of Theoretical Probability)

   Fill, James Allen (Ed.)

   In this paper, we give sufficient conditions for the almost sure central limit theorem started at a point, known under the name of quenched central limit theorem. This is achieved by using a new idea of conditioning with respect to both the past and the future of the Markov chain. As applications, we provide a new sufficient projective condition for the quenched CLT. 

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5. Quantitative finiteness of hyperplanes in hybrid manifolds

   Ohm, Ko; Sanchez, Anthony (June 2025, arXiv)

   We prove a quantitative finiteness theorem for the number of totally geodesic hyperplanes of non-arithmetic hyperbolic n-manifolds that arise from a gluing construction of Gromov and Piatetski-Shapiro for n ≥ 3. This extends work of LindenstraussMohammadi in dimension 3. This follows from effective density theorem for periodic orbits of SO(n −1,1) acting on quotients of SO(n,1) by a lattice for n ≥ 3. The effective density result uses a number of a ideas including Margulis functions, a restricted projection theorem, and an effective equidistribution result for measures on the horospherical subgroup that are nearly full dimensional. 

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