Consider a population evolving as a critical continuous-time Galton–Watson (GW) tree. Conditional on the population surviving until a large time T, sample k individuals uniformly at random (without replacement) from amongst those alive at time T. What is the genealogy of this sample of individuals? In cases where the offspring distribution has finite variance, the probabilistic properties of the joint ancestry of these k particles are well understood, as seen in (Ann. Appl. Probab. 30 (2020) 1368–1414; Electron. J. Probab. 24 (2019) 1–35). In the present article, we study the joint ancestry of a sample of k particles under the following regime: the offspring distribution has mean 1 (critical) and the tails of the offspring distribution are heavy in that $\mathit{\alpha }\in (1,2]$ is the supremum over indices β such that the βth moment is finite. We show that for each α, after rescaling time by $1/\mathit{T}$, there is a universal stochastic process describing the joint coalescent structure of the k distinct particles. The special case $\mathit{\alpha }=2$ generalises the known case of sampling from critical GW trees with finite variance where only pairwise mergers are observed and the genealogical tree is, roughly speaking, some kind of mixture of time-changed Kingman coalescents. The cases $\mathit{\alpha }\in (1,2)$ introduce new universal limiting partition-valued stochastic processes with interesting probabilistic structures, which, in particular, have representations connected to the Lauricella function and the Dirichlet distribution and whose coalescent structures exhibit multiple-mergers of family lines. Moreover, in the case $\mathit{\alpha }\in (1,2)$, we show that the coalescent events of the ancestry of the k particles are associated with birth events that produce giant numbers of offspring of the same order of magnitude as the entire population size, and we compute the joint law of the ancestry together with the sizes of these giant births.

We consider the problem of recovering the community structure in the stochastic block model with two communities. We aim to describe the mutual information between the observed network and the actual community structure in the sparse regime, where the total number of nodes diverges while the average degree of a given node remains bounded. Our main contributions are a conjecture for the limit of this quantity, which we express in terms of a Hamilton–Jacobi equation posed over a space of probability measures, and a proof that this conjectured limit provides a lower bound for the asymptotic mutual information. The well-posedness of the Hamilton–Jacobi equation is obtained in our companion paper. In the case when links across communities are more likely than links within communities, the asymptotic mutual information is known to be given by a variational formula. We also show that our conjectured limit coincides with this formula in this case.

We consider a continuous-time random walk on a regular tree of finite depth and study its favorite points among the leaf vertices. For the walk started from a leaf vertex and stopped upon hitting the root, we prove that, in the limit as the depth of the tree tends to infinity, the suitably scaled and centered maximal time spent at any leaf converges to a randomly-shifted Gumbel law. The random shift is characterized using a derivative-martingale like object associated with square-root local-time process on the tree.

We show that the Weil–Petersson probability that a random surface has first eigenvalue of the Laplacian less than $3/16-\mathit{ϵ}$ goes to zero as the genus goes to infinity.

In this paper we study singular kinetic equations on ${\mathbb{R}}^{2\mathit{d}}$ by the paracontrolled distribution method introduced in Gubinelli, Imkeller and Perkowski (Forum Math. Pi 3 (2015) e6–75). We first develop paracontrolled calculus in the kinetic setting and use it to establish the global well-posedness for the linear singular kinetic equations under the assumptions that the products of singular terms are well defined. We also demonstrate how the required products can be defined in the case that singular term is a Gaussian random field by probabilistic calculation. Interestingly, although the terms in the zeroth Wiener chaos of regularization approximation are not zero, they converge in suitable weighted Besov spaces, and no renormalization is required. As applications the global well-posedness for a nonlinear kinetic equation with singular coefficients is obtained by the entropy method. Moreover, we also solve the martingale problem for nonlinear kinetic distribution dependent stochastic differential equations with singular drifts.

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 (Invent. Math. 223 (2021) 213–333). As a consequence, we prove that the metric structure of a γ-LQG surface determines its conformal structure for every $\mathit{\gamma }\in (0,2)$. Our primary tool is the continuum mating-of-trees 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 consider the Metropolis biased card shuffling (also called the multi-species ASEP on a finite interval or the random Metropolis scan). Its convergence to stationarity was believed to exhibit a total-variation cutoff, and that was proved a few years ago by Labbé and Lacoin (Ann. Probab. 47 (2019) 1541–1586). In this paper, we prove that (for N cards) the cutoff window is in the order of ${\mathit{N}}^{1/3}$, and the cutoff profile is given by the Tracy–Widom GOE distribution function. This confirms a conjecture by Bufetov and Nejjar (Probab. Theory Related Fields 83 (2022) 229–253). Our approach is different from (Ann. Probab. 47 (2019) 1541–1586), by comparing the card shuffling with the multispecies ASEP on $\mathbb{Z}$, and using Hecke algebra and recent ASEP shift-invariance and convergence results. Our result can also be viewed as a generalization of the Oriented Swap Process finishing time convergence (Ann. Appl. Probab. 32 (2022) 753–763), which is the TASEP version (of our result).

Let $\mathit{B}={({\mathit{B}}_{\mathit{x}})}_{\mathit{x}\in {\mathbb{R}}^{\mathit{d}}}$ be a collection of $\mathit{N}(0,1)$ random variables forming a real-valued continuous stationary Gaussian field on ${\mathbb{R}}^{\mathit{d}}$, and set $\mathit{C}(\mathit{x}-\mathit{y})=\mathbb{E}[{\mathit{B}}_{\mathit{x}}{\mathit{B}}_{\mathit{y}}]$. Let $\mathit{\phi }:\mathbb{R}\to \mathbb{R}$ be such that $\mathbb{E}[\mathit{\phi }{(\mathit{N})}^2]<\infty$ with $\mathit{N}\sim \mathit{N}(0,1)$, let R be the Hermite rank of φ, and consider ${\mathit{Y}}_{\mathit{t}}={\int }_{\mathit{t}\mathit{D}}\mathit{\phi }({\mathit{B}}_{\mathit{x}})\mathit{d}\mathit{x}$, $\mathit{t}>0$ with $\mathit{D}\subset {\mathbb{R}}^{\mathit{d}}$ compact.

Since the pioneering works from the 1980s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and noncentral limit theorems for ${\mathit{Y}}_{\mathit{t}}$ 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 $\mathit{t}\to \infty$ the fluctuations of ${\mathit{Y}}_{\mathit{t}}$ 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 $\mathit{R}\ge 2$.

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 ${\mathit{Y}}_{\mathit{t}}$ 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 $\mathit{C}\notin {\mathit{L}}^{\mathit{R}}({\mathbb{R}}^{\mathit{d}})$. Our main mathematical tools are the Malliavin–Stein method and Fourier analysis techniques.