We consider a particle with position (X t ) t≥0 living in R + , whose velocity (V t ) t≥0 is a positive recurrent diffusion with heavy-tailed invariant distribution when the particle lives in (0, ∞). When it hits the boundary x = 0, the particle restarts with a random strictly positive velocity. We show that the properly rescaled position process converges weakly to a stable process reflected on its infimum. From a P.D.E. point of view, the time-marginals of (X t , V t ) t≥0 solve a kinetic Fokker-Planck equation on (0, ∞)×R + ×R with diffusive boundary conditions. Properly rescaled, the space-marginal converges to the solution of some fractional heat equation on (0, ∞) × R + .

The Discrete Gaussian model is the lattice Gaussian free field conditioned to be integervalued. In two dimensions, at sufficiently high temperature, we show that its macroscopic scaling limit on the torus is a multiple of the Gaussian free field. Our proof starts from a single renormalisation group step after which the integer-valued field becomes a smooth field which we then analyse using the renormalisation group method. This paper also provides the foundation for the construction of the scaling limit of the infinite-volume gradient Gibbs state of the Discrete Gaussian model in the companion paper. Moreover, we develop all estimates for general finite-range interaction with sharp dependence on the range. We expect these estimates to prepare for a future analysis of the spread-out version of the Discrete Gaussian model at its critical temperature.

We establish the sharpness of the phase transition for a wide class of Gaussian percolation models, on Zd or Rd, d≥2, with correlations decaying at least algebraically with exponent α>0, including the discrete Gaussian free field (d≥3, α=d−2), the discrete Gaussian membrane model (d≥5, α=d−4), and many other examples both discrete and continuous. In particular, we do not assume positive correlations. This result is new for all strongly correlated models (i.e., α∈(0,d]) in dimension d≥3 except the Gaussian free field, for which sharpness was proven in a recent breakthrough by Duminil-Copin et al. (Duke Math. J. 172 (2023) 839–913); even then, our proof is simpler and yields new near-critical information on the percolation density. For planar fields which are continuous and positively correlated, we establish sharper bounds on the percolation density by exploiting a new ‘weak mixing’ property for strongly correlated Gaussian fields. As a byproduct, we establish the box-crossing property for the nodal set, of independent interest. This is the second in a series of two papers studying level-set percolation of strongly correlated Gaussian fields, which can be read independently.

We establish, in full generality, an unexpected phenomenon of strong regularization along normal convergence on Wiener chaoses. Namely, for every sequence of chaotic random variables, convergence in law to the Gaussian distribution is automatically upgraded to superconvergence: the regularity of the densities increases along the convergence, and all the derivatives converge uniformly on the real line. Our findings strikingly strengthen known results regarding modes of convergence for normal approximation on Wiener chaoses. Without additional assumptions, quantitative convergence in total variation is established by Nourdin and Peccati (Probab. Theory Related Fields 145 (2009) 75–118), and later on amplified to convergence in relative entropy by Nourdin, Peccati and Swan (J. Funct. Anal. 266 (2014) 3170–3207). Our result is then extended to the multivariate setting and for polynomial mappings of a Gaussian field, provided the projection on the Wiener chaos of maximal degree admits a nondegenerate Gaussian limit. While our findings potentially apply to any context involving polynomial functionals of a Gaussian field, we emphasize, in this work, applications regarding: improved Carbery–Wright estimates near Gaussianity, normal convergence in entropy and in Fisher information, superconvergence for the spectral moments of Gaussian orthogonal ensembles, moments bounds for the inverse of strongly correlated Wishart-type matrices, and superconvergence in the Breuer–Major Theorem. Our proofs leverage Malliavin’s historical idea to establish smoothness of the density via the existence of negative moments of the Malliavin gradient, and we further develop a new paradigm to study this problem. Namely, we relate the existence of negative moments to some explicit spectral quantities associated with the Malliavin Hessian. This link relies on an adequate choice of the Malliavin gradient, which provides a novel decoupling procedure of independent interest. Previous attempts to establish convergence beyond entropy have imposed restrictive assumptions ensuring finiteness of negative moments for the Malliavin derivatives Our analysis renders these assumptions superfluous. The terminology superconvergence was introduced by Bercovici and Voiculescu (Probab. Theory Related Fields 103 (1995) 215–222) for the central limit theorem in free probability.

Let (Formula presented) be a collection of N(0, 1) random variables forming a real-valued continuous stationary Gaussian field on (Formula presented), and set (Formula presented). Let (Formula presented) be such that (Formula presented), let R be the Hermite rank of φ, and consider (Formula presented) with (Formula presented) compact. Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for Yt have been constantly refined, extended and applied to an increasing number of diverse situations, to such an extent that it has become a field of research in its own right. The common belief, representing the intuition that specialists in the subject have developed over the last four decades, is that as t→∞the fluctuations of Yt around its mean are, in general (i.e., except possibly in very special cases), Gaussian when B has short memory, and non-Gaussian when B has long memory and R ≥ 2. We show in this paper that this intuition forged over the last 40 years can be wrong, and not only marginally or in critical cases. We will indeed bring to light a variety of situations where Yt admits Gaussian fluctuations in a long memory context. To achieve this goal, we state and prove a spectral central limit theorem, which extends the conclusion of the celebrated Breuer-Major theorem to situations where (Formula presented) Our main mathematical tools are the Malliavin- Stein method and Fourier analysis techniques.

A Liouville quantum gravity (LQG) surface is a natural random two-dimensional surface, initially formulated as a random measure space and later as a random metric space. We show that the LQG measure can be recovered as the Minkowski measure with respect to the LQG metric, answering a question of Gwynne and Miller [GM21c]. As a consequence, we prove that the metric structure of a γ-LQG surface determines its conformal structure for every γ ∈ (0, 2). Our primary tool is the continuum mating-oftrees theory for space-filling SLE. In the course of our proof, we also establish a Hölder continuity result for space-filling SLE with respect to the LQG metric.

We establish central limit theorems for the volumes of intersections of Bpn (the unit ball of ℓpn) with uniform random subspaces of codimension d for fixed d and n→∞. As a corollary we obtain higher-order approximations for expected volumes, refining previous results by Koldobsky and Lifschitz and approximations obtained from the Eldan–Klartag version of CLT for convex bodies. We also obtain a central limit theorem for the Minkowski functional of the intersection body of Bpn, evaluated on a random vector distributed uniformly on the unit sphere.

Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations.

Place an A-particle at each site of a graph independently with probability p, and otherwise place a B-particle. A- and B-particles perform independent continuous time random walks at rates λA and λB, respectively, and annihilate upon colliding with a particle of opposite type. Bramson and Lebowitz studied the setting λA=λB in the early 1990s. Despite recent progress, many basic questions remain unanswered when λA≠λB. For the critical case p=1/2 on low-dimensional integer lattices, we give a lower bound on the expected number of particles at the origin that matches physicists’ predictions. For the process with λB=0 on the integers and on the bidirected regular tree, we give sharp upper and lower bounds for the expected total occupation time of the root at and approaching criticality.

We consider the N -particle Fleming-Viot process associated to a normally reflected diffusion with soft catalyst killing. The Fleming-Viot multicolor process is obtained by attaching genetic information to the particles in the Fleming-Viot process. We establish that, after rescaling time by t → Nt, this genetic information converges to the (very different) Fleming-Viot process from population genetics, as N → ∞. An extension is provided to dynamics given by Brownian motion with hard catalyst killing at the boundary of its domain.