We establish a new scale of p-variation estimates for martingale paraproducts, martingale transforms and Itô integrals, of relevance in rough paths theory, stochastic and harmonic analysis. As an application, we introduce rough semimartingales, a common generalization of classical semimartingales and (controlled) rough paths and their integration theory.

The main results of the article are short time estimates and asymptotic estimates for the first two order derivatives of the logarithmic heat kernel of a complete Riemannian manifold. We remove all curvature restrictions and also develop several techniques.

A basic tool developed here is intrinsic stochastic variations with prescribed second order covariant differentials, allowing to obtain a path integration representation for the second order derivatives of the heat semigroup ${\mathit{P}}_{\mathit{t}}$ on a complete Riemannian manifold, again without any assumptions on the curvature. The novelty is the introduction of an ${\mathit{ϵ}}^2$ term in the variation allowing greater control. We also construct a family of cut-off stochastic processes adapted to an exhaustion by compact subsets with smooth boundaries, each process is constructed path by path and differentiable in time. Furthermore, the differentials have locally uniformly bounded moments with respect to the Brownian motion measures, allowing to bypass the lack of continuity of the exit time of the Brownian motions on its initial position.

We consider the exclusion process with reservoirs on arbitrary networks. We characterize the spectral gap, mixing time, and mixing window of the process, in terms of certain simple spectral statistics of the underlying network. Among other consequences we establish a nonconservative analogue of Aldous’s spectral gap conjecture, and we show that cutoff occurs if and only if the product condition is satisfied. We illustrate this by providing explicit cutoffs on discrete lattices of arbitrary dimensions and boundary conditions which substantially generalize recent one-dimensional results. We also obtain cutoff phenomena in relative entropy, Hilbert norm, separation distance, and supremum norm. Our proof exploits negative dependence in a novel, simple way to reduce the understanding of the whole process to that of single-site marginals. We believe that this approach will find other applications.

Let ${\mathit{A}}_{\mathit{n}}={\mathit{\epsilon }}_{\mathit{n}}\cdots {\mathit{\epsilon }}_1$, where ${({\mathit{\epsilon }}_{\mathit{n}})}_{\mathit{n}\ge 1}$ is a sequence of independent random matrices, taking values in ${\mathrm{GL}}_{\mathit{d}}(\mathbb{R})$, $\mathit{d}\ge 2$, with common distribution μ. In this paper, under standard assumptions on μ (strong irreducibility and proximality) we prove Berry–Esseen type theorems for $log(\Vert {\mathit{A}}_{\mathit{n}}\Vert )$ when μ has a polynomial moment. More precisely, we get the rate ${((log\mathit{n})/\mathit{n})}^{\mathit{q}/2-1}$, when μ has a moment of order $\mathit{q}\in ]2,3]$ and the rate $1/\sqrt{\mathit{n}}$ when μ has a moment of order 4, which significantly improves earlier results in this setting.

We are concerned with the three-dimensional incompressible Navier–Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in ${\mathit{L}}^2$ we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in particular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.

We consider percolation on ${\mathbb{Z}}^{\mathit{d}}$ and on the d-dimensional discrete torus, in dimensions $\mathit{d}\ge 11$ for the nearest-neighbour model and in dimensions $\mathit{d}>6$ for spread-out models. For ${\mathbb{Z}}^{\mathit{d}}$ we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy

Using this, we prove that throughout the critical window the torus two-point function has a “plateau,” meaning that it decays for small x as $\Vert \mathit{x}{\Vert }^{-(\mathit{d}-2)}$ but for large x is essentially constant and of order ${\mathit{V}}^{-2/3}$ where V is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on ${\mathbb{Z}}^{\mathit{d}}$ are separated by a multiple of ${\mathit{V}}^{-1/3}$. The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the ${\mathbb{Z}}^{\mathit{d}}$ two-point function. In particular, we use results derived from the lace expansion on ${\mathbb{Z}}^{\mathit{d}}$, but in contrast to previous work on high-dimensional torus percolation, we do not need or use a separate torus lace expansion.

We characterize the behavior of a random discrete interface ϕ on ${[-\mathit{L},\mathit{L}]}^{\mathit{d}}\cap {\mathbb{Z}}^{\mathit{d}}$ with energy $\sum \mathit{V}(\mathrm{\Delta }\mathit{\varphi }(\mathit{x}))$ as $\mathit{L}\to \infty$, where Δ is the discrete Laplacian and V is a uniformly convex, symmetric, and smooth potential. The interface ϕ is called the non-Gaussian membrane model. By analyzing the Helffer–Sjöstrand representation, associated to $\mathrm{\Delta }\mathit{\varphi }$, we provide a unified approach to continuous scaling limits of the rescaled and interpolated interface in dimensions $\mathit{d}=2,3$, Gaussian approximation in negative regularity spaces for all $\mathit{d}\ge 2$, and the infinite volume limit in $\mathit{d}\ge 5$.

We show that the naive mean-field approximation correctly predicts the leading term of the logarithmic lower tail probabilities for the number of copies of a given subgraph in $\mathit{G}(\mathit{n},\mathit{p})$ and of arithmetic progressions of a given length in random subsets of the integers in the entire range of densities where the mean-field approximation is viable.

Our main technical result provides sufficient conditions on the maximum degrees of a uniform hypergraph $\mathcal{H}$ that guarantee that the logarithmic lower tail probabilities for the number of edges, induced by a binomial random subset of the vertices of $\mathcal{H}$, can be well approximated by considering only product distributions. This may be interpreted as a weak, probabilistic version of the hypergraph container lemma that is applicable to all sparser-than-average (and not only independent) sets.

We study the stability of entropically regularized optimal transport with respect to the marginals. Given marginals converging weakly, we establish a strong convergence for the Schrödinger potentials, describing the density of the optimal couplings. When the marginals converge in total variation, the optimal couplings also converge in total variation. This is applied to show that Sinkhorn’s algorithm converges in total variation when costs are quadratic and marginals are subgaussian or, more generally, for all continuous costs satisfying an integrability condition.

In the seventies, Charles Stein revolutionized the way of proving the central limit theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last 50 years, much research has been done to adapt and strengthen this method to a variety of different settings and other limiting distributions. However, it has not been yet extended to study conditional convergences. In this article we develop a novel approach, using Stein’s method for exchangeable pairs, to find a rate of convergence in the conditional central limit theorem of the form $({\mathit{X}}_{\mathit{n}}\mid {\mathit{Y}}_{\mathit{n}}=\mathit{k})$, where $({\mathit{X}}_{\mathit{n}},{\mathit{Y}}_{\mathit{n}})$ are asymptotically jointly Gaussian, and extend this result to a multivariate version. We apply our general result to several concrete examples, including pattern count in a random binary sequence and subgraph count in Erdős–Rényi random graph.

The Tracy–Widom distributions are among the most famous laws in probability theory, partly due to their connection with Wigner matrices. In particular, for $\mathit{A}=\frac{1}{\sqrt{\mathit{n}}}{({\mathit{a}}_{\mathit{i}\mathit{j}})}_{1\le \mathit{i},\mathit{j}\le \mathit{n}}\in {\mathbb{R}}^{\mathit{n}\times \mathit{n}}$ symmetric with ${({\mathit{a}}_{\mathit{i}\mathit{j}})}_{1\le \mathit{i}\le \mathit{j}\le \mathit{n}}$ i.i.d. standard normal, the fluctuations of its largest eigenvalue ${\mathit{\lambda }}_1(\mathit{A})$ are asymptotically described by a real-valued Tracy–Widom distribution ${\mathit{TW}}_1:{\mathit{n}}^{2/3}({\mathit{\lambda }}_1(\mathit{A})-2)\Rightarrow {\mathit{TW}}_1$. As it often happens, Gaussianity can be relaxed, and this results holds when $\mathbb{E}[{\mathit{a}}_{11}]=0$, $\mathbb{E}[{\mathit{a}}_{11}^2]=1$ and the tail of ${\mathit{a}}_{11}$ decays sufficiently fast: ${lim}_{\mathit{x}\to \infty }{\mathit{x}}^4\mathbb{P}(|{\mathit{a}}_{11}|>\mathit{x})=0$, whereas when the law of ${\mathit{a}}_{11}$ is regularly varying with index $\mathit{\alpha }\in (0,4)$, ${\mathit{c}}_{\mathit{a}}(\mathit{n}){\mathit{n}}^{1/2-2/\mathit{\alpha }}{\mathit{\lambda }}_1(\mathit{A})$ converges to a Fréchet distribution for ${\mathit{c}}_{\mathit{a}}:(0,\infty )\to (0,\infty )$, slowly varying and depending solely on the law of ${\mathit{a}}_{11}$. This paper considers a family of edge cases, ${lim}_{\mathit{x}\to \infty }{\mathit{x}}^4\mathbb{P}(|{\mathit{a}}_{11}|>\mathit{x})=\mathit{c}\in (0,\infty )$, and unveils a new type of limiting behavior for ${\mathit{\lambda }}_1(\mathit{A})$: a continuous function of a Fréchet distribution in which 2, the almost sure limit of ${\mathit{\lambda }}_1(\mathit{A})$ in the light-tailed case, plays a pivotal role: