We quantify the behaviour at large scales of the beta coalescent $\mathrm{\Pi }=\{\mathrm{\Pi }(\mathit{t}),\mathit{t}\ge 0\}$ with parameters $\mathit{a},\mathit{b}>0$. Specifically, we study the rescaled block size spectrum of $\mathrm{\Pi }(\mathit{t})$ and of its restriction ${\mathrm{\Pi }}_{\mathit{n}}(\mathit{t})$ to $\{1,\dots ,\mathit{n}\}$. Our main result is a law of large numbers type of result if Π comes down from infinity. In the case of Kingman’s coalescent the derivation of this so-called hydrodynamic limit has been known since the work of Smoluchowski (Z. Phys. 17 (1916) 557–585). We extend Smoluchowski’s result to beta coalescents and show that if Π comes down from infinity both rescaled spectra

$${\mathit{n}}^{-1}({\mathfrak{c}}_1\mathrm{\Pi }(\mathit{t}{\mathit{\tau }}_{\mathit{n}}),\dots ,{\mathfrak{c}}_{\mathit{n}}\mathrm{\Pi }(\mathit{t}{\mathit{\tau }}_{\mathit{n}})),\text{and}{\mathit{n}}^{-1}({\mathfrak{c}}_1{\mathrm{\Pi }}_{\mathit{n}}(\mathit{t}{\mathit{\tau }}_{\mathit{n}}),\dots ,{\mathfrak{c}}_{\mathit{n}}{\mathrm{\Pi }}_{\mathit{n}}(\mathit{t}{\mathit{\tau }}_{\mathit{n}})),$$

converge to (different) deterministic limits that we compute explicitly in terms of partial Bell polynomials. Here ${\mathfrak{c}}_{\mathit{i}}\mathit{\pi }$ counts the number of blocks of size i in a partition π, and $({\mathit{\tau }}_{\mathit{n}})$ is a sequence such that ${\mathit{\tau }}_{\mathit{n}}\sim {\mathit{n}}^{-(1-\mathit{a})}$ as $\mathit{n}\to \infty$.

Along the way we study the nontrivial limits of the rescaled block counting processes $\{{\mathit{n}}^{\mathit{\alpha }}\mathrm{\#}{\mathrm{\Pi }}_{\mathit{n}}(\mathit{t}{\mathit{\tau }}_{\mathit{n}}),\mathit{t}\ge 0\}$, and $\{{\mathit{n}}^{\mathit{\alpha }}\mathrm{\#}\mathrm{\Pi }(\mathit{t}{\mathit{\tau }}_{\mathit{n}}),\mathit{t}\ge 0\}$, where $\mathit{\alpha }\in [-1,-2/(3-\mathit{a}))$, and ${\mathit{\tau }}_{\mathit{n}}\sim {\mathit{n}}^{\mathit{\alpha }(1-\mathit{a})}$ if Π comes down from infinity.

This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized, have been successfully applied in a wide range of static and dynamic learning problems. Despite the popularity of this approach in empirical tasks, important theoretical questions regarding the relation between the unknown function, the weight distribution, and the approximation rate have remained open. In this work it is proved that, as long as the unknown function, functional, or dynamical system is sufficiently regular, it is possible to draw the internal weights of the random (recurrent) neural network from a generic distribution (not depending on the unknown object) and quantify the error in terms of the number of neurons and the hyperparameters. In particular, this proves that echo state networks with randomly generated weights are capable of approximating a wide class of dynamical systems arbitrarily well and thus provides the first mathematical explanation for their empirically observed success at learning dynamical systems.

The convergence rate in Wasserstein distance is estimated for the empirical measures of symmetric semilinear SPDEs. Unlike in the finite-dimensional case that the convergence is of algebraic order in time, in the present situation the convergence is of log order with a power given by eigenvalues of the underlying linear operator.

Fix $\mathit{t}\ge 2$. We first give an asymptotic formula for certain sums of the number of t-cores. We then use this result to compute the distribution of the size of the t-core of a uniformly random partition of an integer n. We show that this converges weakly to a gamma distribution after dividing by $\sqrt{\mathit{n}}$. As a consequence, we find that the size of the t-core is of the order of $\sqrt{\mathit{n}}$ in expectation. We then apply this result to show that the probability that t divides the hook length of a uniformly random cell in a uniformly random partition equals $1/\mathit{t}$ in the limit. Finally, we extend this result to all modulo classes of t using abacus representations for cores and quotients.

We introduce and study a new percolation model, inspired by recent works on jigsaw percolation, graph bootstrap percolation, and percolation in polluted environments. Start with an oriented graph ${\mathit{G}}_0$ of initially occupied edges on n vertices, and iteratively occupy additional (oriented) edges by transitivity, with the constraint that only open edges in a certain random set can ever be occupied. All other edges are closed, creating a set of obstacles for the spread of occupied edges. When ${\mathit{G}}_0$ is an unoriented linear graph, and leftward and rightward edges are open independently with possibly different probabilities, we identify three regimes in which the set of eventually occupied edges is either all open edges, the majority of open edges in one direction, or only a very small proportion of all open edges. In the more general setting where ${\mathit{G}}_0$ is a connected unoriented graph of bounded degree, we show that the transition between sparse and full occupation of open edges occurs when the probability of open edges is ${(log\mathit{n})}^{-1/2+\mathit{o}(1)}$. We conclude with several conjectures and open problems.

Let $\mathit{S}(\mathit{n})$ be a centered random walk with finite second moment. We consider the integrated random walk $\mathit{T}(\mathit{n})=\mathit{S}(0)+\mathit{S}(1)+\cdots +\mathit{S}(\mathit{n})$. We prove invariance principles for the meander and for the bridge of this process, under the condition that the integrated random walk remains positive. Furthermore, we prove the functional convergence of its Doob’s h-transform to the h-transform of the Kolmogorov diffusion conditioned to stay positive.

Given a sequence of i.i.d. random functions ${\mathrm{\Psi }}_{\mathit{n}}:\mathbb{R}\to \mathbb{R}$, $\mathit{n}\in \mathbb{N}$, we consider the iterated function system and Markov chain, which is recursively defined by ${\mathit{X}}_0^{\mathit{x}}:=\mathit{x}$ and ${\mathit{X}}_{\mathit{n}}^{\mathit{x}}:={\mathrm{\Psi }}_{\mathit{n}-1}({\mathit{X}}_{\mathit{n}-1}^{\mathit{x}})$ for $\mathit{x}\in \mathbb{R}$ and $\mathit{n}\in \mathbb{N}$. Under the two basic assumptions that the ${\mathrm{\Psi }}_{\mathit{n}}$ are a.s. continuous at any point in $\mathbb{R}$ and asymptotically linear at the “endpoints” $\pm \infty$, we study the tail behavior of the stationary laws of such Markov chains by means of Markov renewal theory. Our approach provides an extension of Goldie’s implicit renewal theory (Ann. Appl. Probab. (1991) 1 126–166) and can also be viewed as an adaptation of Kesten’s work on products of random matrices (Acta Math. (1973) 131 207–248) to one-dimensional function systems as described. Our results have applications in quite different areas of applied probability like queuing theory, econometrics, mathematical finance and population dynamics, for example, $\mathsf{ARCH}$ models and random logistic transforms.

With the recent developments on nonlinear SPDEs, where smoothing of rough noises is needed, one is naturally led to study interacting particle systems whose macroscopic evolution is described by these equations and which possess an in-built smoothing. In this article, our main results are to derive regularized versions of the ill-posed one-dimensional SPDE

$${\partial }_{\mathit{t}}\mathit{\rho }=\frac{1}{2}\mathrm{\Delta }\mathrm{\Phi }(\mathit{\rho })-2\nabla ({\mathit{W}}^{\prime }\mathrm{\Phi }(\mathit{\rho })),$$

where the spatial white noise ${\mathit{W}}^{\prime }$ is replaced by a regularization ${\mathit{W}}_{\mathit{\epsilon }}^{\prime }$, as quenched and annealed hydrodynamic limits of zero-range interacting particle systems in ε-regularized Sinai-type random environments. Some computations are also made about annealed mean hydrodynamic limits in unregularized Sinai-type random environments with respect to independent particles.

We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process X. We consider classic and randomized stopping times represented by linear and nonlinear functionals of the rough path signature ${\mathbb{X}}^{<\infty }$ associated to X, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature $\mathbb{E}[{\mathbb{X}}_{0,\mathit{T}}^{\le \mathit{N}}]$. By applying a deep neural network approach to approximate the nonlinear signature functionals, we can efficiently solve the optimal stopping problem numerically. The only assumption on the process X is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semimartingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, for example, on financial or electricity markets.

The supercooled Stefan problem and its variants describe the freezing of a supercooled liquid in physics, as well as the large system limits of systemic risk models in finance and of integrate-and-fire models in neuroscience. Adopting the physics terminology, the supercooled Stefan problem is known to feature a finite-time blow-up of the freezing rate for a wide range of initial temperature distributions in the liquid. Such a blow-up can result in a discontinuity of the liquid-solid boundary. In this paper, we prove that the natural Euler time-stepping scheme applied to a probabilistic formulation of the supercooled Stefan problem converges to the liquid-solid boundary of its physical solution globally in time, in the Skorokhod M1 topology. In the course of the proof, we give an explicit bound on the rate of local convergence for the time-stepping scheme. We also run numerical tests to compare our theoretical results to the practically observed convergence behavior.

Over 80% of human cancers originate from the epithelium, which covers the outer and inner surfaces of organs and blood vessels. In stratified epithelium, the bottom layers are occupied by stem and stem-like cells that continually divide and replenish the upper layers. In this work, we study the spread of premalignant mutant clones and cancer initiation in stratified epithelium, using the biased voter model on stacked two-dimensional lattices. Our main result is an estimate of the propagation speed of a premalignant mutant clone, which is asymptotically precise in the cancer-relevant weak-selection limit. We use our main result to study cancer initiation under a two-step mutational model of cancer, which includes computing the distributions of the time of cancer initiation and the size of the premalignant clone giving rise to cancer. Our work quantifies the effect of epithelial tissue thickness on the process of carcinogenesis, thereby contributing to an emerging understanding of the spatial evolutionary dynamics of cancer.

We prove that stochastic replicator dynamics can be interpreted as intrinsic Brownian motion on the simplex equipped with the Aitchison geometry. As an immediate consequence, we derive three approximation results in the spirit of Wong–Zakai approximation, Donsker’s invariance principle and a JKO-scheme. Using the Fokker–Planck equation and Wasserstein-contraction estimates, we also study the long time behavior of the stochastic replicator equation, as an example of a nongradient drift diffusion on the Aitchison simplex.

We describe and analyze a class of positive recurrent reflected Brownian motions (RBMs) in ${\mathbb{R}}_{+}^{\mathit{d}}$ for which local statistics converge to equilibrium at a rate independent of the dimension d. Under suitable assumptions on the reflection matrix, drift and diffusivity coefficients, dimension-independent stretched exponential convergence rates are obtained by estimating contractions in an underlying weighted distance between synchronously coupled RBMs. We also study the symmetric Atlas model as a first step in obtaining dimension-independent convergence rates for RBMs not satisfying the above assumptions. By analyzing a pathwise derivative process and connecting it to a random walk in a random environment, we obtain polynomial convergence rates for the gap process of the symmetric Atlas model started from appropriate perturbations of stationarity.

We analyze the strong noise limit of one-dimensional stochastic differential equations (SDEs).

Our initial motivation comes from continuous measurements of open quantum systems. In this context, Bauer, Bernard and Tilloy pointed out an intriguing behavior. As the noise grows larger, the solutions exhibit locally a collapsing, that is to say, converge to pure jump processes very reminiscent of a metastability phenomenon. But surprisingly the limiting jump process is decorated by a spike process.

We give a precise meaning to the convergence and completely prove these statements for a large class of one-dimensional diffusions, thanks to a robust strategy of proof.

We consider the fluctuations of regular functions f of a Wigner matrix W viewed as an entire matrix $\mathit{f}(\mathit{W})$. Going beyond the well-studied tracial mode, $Tr\mathit{f}(\mathit{W})$, which is equivalent to the customary linear statistics of eigenvalues, we show that $Tr\mathit{f}(\mathit{W})\mathit{A}$ is asymptotically normal for any nontrivial bounded deterministic matrix A. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of $\mathit{f}(\mathit{W})$ in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. As a main motivation to study CLT in such generality on small mesoscopic scales, we determine the fluctuations in the eigenstate thermalization hypothesis (Phys. Rev. A 43 (1991) 2046–2049), that is, prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. Finally, in the macroscopic regime our result also generalizes (Zh. Mat. Fiz. Anal. Geom. 9 (2013) 536–581, 611, 615) to complex W and to all crossover ensembles in between. The main technical inputs are the recent multiresolvent local laws with traceless deterministic matrices from the companion paper (Comm. Math. Phys. 388 (2021) 1005–1048).

In the present work, we investigate the case of directed polymer in a random environment (DPRE), when the increments of the one-dimensional random walk are heavy-tailed with tail-exponent equal to zero ($\mathbf{P}[|{\mathit{X}}_1|\ge \mathit{n}]$ decays slower than any power of n). This case has not yet been studied in the context of directed polymers and presents key differences with the simple symmetric random walk case and the cases where the increments belong to the domain of attraction of an α-stable law, where $\mathit{\alpha }\in (0,2]$. We establish the absence of a very strong disorder regime—that is, the free energy equals zero at every temperature—for every disorder distribution. We also prove that a strong disorder regime (partition function converging to zero at low temperature) may exist or not depending on finer properties of the random walk: we establish nonmatching necessary and sufficient conditions for having a phase transition from weak to strong disorder. In particular our results imply that for this directed polymer model, very strong disorder is not equivalent to strong disorder, shedding a new light on a long standing conjecture concerning the original nearest-neighbor DPRE.

We employ stabilization methods and second order Poincaré inequalities to establish rates of multivariate normal convergence for a large class of vectors $({\mathit{H}}_{\mathit{s}}^{(1)},\dots ,{\mathit{H}}_{\mathit{s}}^{(\mathit{m})})$, $\mathit{s}\ge 1$, of statistics of marked Poisson processes on ${\mathbb{R}}^{\mathit{d}}$, $\mathit{d}\ge 2$, as the intensity parameter s tends to infinity. Our results are applicable whenever the functionals ${\mathit{H}}_{\mathit{s}}^{(\mathit{i})}$, $\mathit{i}\in \{1,\dots ,\mathit{m}\}$, are expressible as sums of exponentially stabilizing score functions satisfying a moment condition. The rates are for the ${\mathit{d}}_2$-, ${\mathit{d}}_3$-, and ${\mathit{d}}_{\mathrm{convex}}$-distances and are in general unimprovable. When we compare with a centered Gaussian random vector, whose covariance matrix is given by the asymptotic covariances, the rates are governed by the rate of convergence of ${\mathit{s}}^{-1}Cov({\mathit{H}}_{\mathit{s}}^{(\mathit{i})},{\mathit{H}}_{\mathit{s}}^{(\mathit{j})})$, $\mathit{i},\mathit{j}\in \{1,\dots ,\mathit{m}\}$, to the limiting covariance, shown to be at most of order ${\mathit{s}}^{-1/\mathit{d}}$. We use the general results to deduce rates of multivariate normal convergence for statistics arising in random graphs and topological data analysis as well as for multivariate statistics used to test equality of distributions. Some of our results hold for stabilizing functionals of Poisson input on suitable metric spaces.

We investigate the characteristic polynomials ${\mathit{\phi }}_{\mathit{N}}$ of the Gaussian β-ensemble for general $\mathit{\beta }>0$ through its transfer matrix recurrence. Our motivation is to obtain a (probabilistic) approximation for ${\mathit{\phi }}_{\mathit{N}}$ in terms of a Gaussian log-correlated field. We distinguish between different types of transfer matrices and analyze completely the hyperbolic part of the recurrence. As a result, we obtain a new coupling between ${\mathit{\phi }}_{\mathit{N}}$ and a Gaussian analytic function with an error which is uniform away from the support of the semicircle law. We use this as input to give the almost sure scaling limit of the characteristic polynomial at the edge in (Lambert and Paquette (2020)). This is also required to obtain analogous strong approximations inside of the bulk of the semicircle law. Our analysis relies on moderate deviation estimates for the product of transfer matrices and this approach might also be useful in different contexts.

Determinantal point processes (DPPs) are a class of repulsive point processes, popular for their relative simplicity. They are traditionally defined via their marginal distributions, but a subset of DPPs called “L-ensembles” have tractable likelihoods and are thus particularly easy to work with. Indeed, in many applications, DPPs are more naturally defined based on the L-ensemble formulation rather than through the marginal kernel.

The fact that not all DPPs are L-ensembles is unfortunate, but there is a unifying description. We introduce here extended L-ensembles, and show that all DPPs are extended L-ensembles (and vice versa). Extended L-ensembles have very simple likelihood functions, contain L-ensembles and projection DPPs as special cases. From a theoretical standpoint, they fix some pathologies in the usual formalism of DPPs, for instance, the fact that projection DPPs are not L-ensembles. From a practical standpoint, they extend the set of kernel functions that may be used to define DPPs: we show that conditional positive definite kernels are good candidates for defining DPPs, including DPPs that need no spatial scale parameter.

Finally, extended L-ensembles are based on so-called “saddle-point matrices”, and we prove an extension of the Cauchy–Binet theorem for such matrices that may be of independent interest.

We establish cutoff for a natural random walk ($\mathsf{RW}$) on the set of perfect matchings ($\mathsf{PM}$s), based on “rewiring”. An n-$\mathsf{PM}$ is a pairing of $2\mathit{n}$ objects. The k-$\mathsf{PM}$ $\mathsf{RW}$ selects k pairs uniformly at random, disassociates the corresponding $2\mathit{k}$ objects, then chooses a new pairing on these $2\mathit{k}$ objects uniformly at random. The equilibrium distribution is uniform over all n-$\mathsf{PM}$s.

The 2-$\mathsf{PM}$ $\mathsf{RW}$ was first introduced by Diaconis and Holmes (Proc. Natl. Acad. Sci. USA 95 (1998) 14600–14602; Electron. J. Probab. 7 (2002) no. 6), seen as a $\mathsf{RW}$ on phylogenetic trees. They established cutoff in this case. We establish cutoff for the k-$\mathsf{PM}$ $\mathsf{RW}$ whenever $2\le \mathit{k}\ll \mathit{n}$. If $\mathit{k}\gg 1$, then the mixing time is $\frac{\mathit{n}}{\mathit{k}}log\mathit{n}$ to leading order.

Diaconis and Holmes (Electron. J. Probab. 7 (2002) no. 6) relate the 2-$\mathsf{PM}$ $\mathsf{RW}$ to the random transpositions card shuffle. Ceccherini-Silberstein, Scarabotti and Tolli (J. Math. Sci. 141 (2007) 1182–1229; Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains (2008) Cambridge Univ. Press) establish the same result using representation theory. We are the first to handle $\mathit{k}>2$. We relate the $\mathsf{PM}$ $\mathsf{RW}$ to conjugacy-invariant $\mathsf{RW}$s on the permutation group by introducing a “cycle structure” for $\mathsf{PM}$s, then build on work of Berestycki, Schramm, Şengül and Zeitouni (Israel J. Math. 147 (2005) 221–243; Ann. Probab. 39 (2011) 1815–1843; Probab. Theory Related Fields 173 (2019) 1197–1241) on such $\mathsf{RW}$s.

We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix ${\mathit{X}}^{\ast }\mathit{X}$ converge to its Tracy–Widom limit at a rate nearly ${\mathit{N}}^{-1/3}$, where X is an $\mathit{M}\times \mathit{N}$ random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate ${\mathit{N}}^{-2/9}$ obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.

In this paper, we focus on nonasymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (nonconstant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step (usually called unadjusted Langevin algorithm in the Monte Carlo literature) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under uniform ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for total variation and ${\mathit{L}}^1$-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using stochastic analysis adapted to decreasing step setting.

This work develops new results for stochastic approximation algorithms. The emphases are on treating algorithms and limits with discontinuities. The main ingredients include the use of differential inclusions, set-valued analysis, nonsmooth analysis, and stochastic differential inclusions. Under broad conditions, it is shown that a suitably scaled sequence of the iterates has a differential inclusion limit. In addition, it is shown for the first time that a centered and scaled sequence of the iterates converges weakly to a stochastic differential inclusion limit. The results are then used to treat several application examples including Markov decision processes, Lasso algorithms, Pegasos algorithms, support vector machine classification, and learning. Some numerical demonstrations are also provided.