The adapted weak topology is an extension of the weak topology for stochastic processes designed to adequately capture properties of underlying filtrations. With the recent work of Bart–Beiglböck–P. [7] as starting point, the purpose of this note is to recover with topological arguments the intriguing result by Backhoff–Bartl–Beiglböck–Eder [3] that all adapted topologies in discrete time coincide. We also derive new characterizations of this topology, including descriptions of its trace on the sets of Markov processes and processes equipped with their natural filtration. To emphasize the generality of the argument, we also describe the classical weak topology for measures on ${\mathbb{R}}^d$ by a weak Wasserstein metric based on the theory of weak optimal transport that was initiated by Gozlan–Roberto–Samson–Tetali [11].

This paper considers the $(n,k)$-Bernoulli–Laplace urn model in the case when there are two urns containing n balls each, with two different colors of balls (red and white). In our setting, the total number of red and white balls is the same. Our focus is on the large-time behavior of the corresponding Markov chain tracking the number of red balls in a given urn assuming that the number of selections k at each step obeys $\mathit{\alpha }\le k∕n\le \mathit{\beta }$, where $\mathit{\alpha },\mathit{\beta }$ are constants satisfying $0<\mathit{\alpha }<\mathit{\beta }<\frac{1}{2}$. Under this assumption, cutoff in the total variation distance is established and a cutoff window is provided. The results in this paper solve an open problem posed by Eskenazis and Nestoridi in [8].

We give a description of the high-dimensional limit of one-pass single-batch stochastic gradient descent (SGD) on a least squares problem. This limit is taken with non-vanishing step-size, and with proportionally related number of samples to problem-dimensionality. The limit is described in terms of a stochastic differential equation in high dimensions, which is shown to approximate the state evolution of SGD. As a corollary, the statistical risk is shown to be approximated by the solution of a convolution-type Volterra equation with vanishing errors as dimensionality tends to infinity. The sense of convergence is the weakest that shows that statistical risks of the two processes coincide. This is distinguished from existing analyses by the type of high-dimensional limit given as well as generality of the covariance structure of the samples.

Bougerol’s identity in law is one of the most infamous in the study of Brownian motion. To that end, in this paper, we present a further extension of its study through a proof in the third dimension. Second-dimension proofs have been extensively explored, as have those of the third dimension. However, here we will extrapolate a previously two-dimensional proof into the third dimension through the use of a three-Brownian joint distribution.

We prove that the solution to the singular-degenerate stochastic fast-diffusion equation with parameter $m\in (0,1)$, with zero Dirichlet boundary conditions on a bounded domain in any spatial dimension, and driven by linear multiplicative Wiener noise, exhibits improved regularity in the Sobolev space $W_0^{1,m+1}$ for initial data in $L^2$.

Motivated by a problem posed by Aldous [2, 1] our goal is to find the maximal-entropy win-martingale:

In a sports game between two teams, the chance the home team wins is initially $x_0\in (0,1)$ and finally 0 or 1. As an idealization we take a continuous time interval $[0,1]$ and let $M_t$ be the probability at time t that the home team wins. Mathematically, $M={(M_t)}_{t\in [0,1]}$ is modelled as a continuous martingale. We consider the problem to find the most random martingale M of this type, where ‘most random’ is interpreted as a maximal entropy criterion. In discrete time this is equivalent to the minimization of relative entropy w.r.t. a Gaussian random walk. The continuous time analogue is that the max-entropy win-martingale M should minimize specific relative entropy with respect to Brownian motion in the sense of Gantert [20]. We use this to prove that M is characterized by the stochastic differential equation

$$d{M}_t=\frac{sin(\mathrm{\pi }{M}_t)}{\mathrm{\pi }\sqrt{1-t}}d{B}_t.$$

To derive the form of the optimizer we use a scaling argument together with a new first order condition for martingale optimal transport, which may be of interest in its own right.

In this paper, we give the moderate deviation principle from the hydrodynamic limit of the simple symmetric exclusion process on the 1-dimensional torus starting from a nonequilibrium state, which extends the result given in Gao and Quastel (2003) about the case where the process starts from an equilibrium state. The exponential tightness of the scaled density field of the process and a replacement lemma play key roles in the proof of the main result. We utilize Grownwall’s inequality and the upper bound of the large deviation principle given in Kipnis, Olla and Varadhan (1989) to prove the above exponential tightness and the replacement lemma respectively in the absence of the invariance of the initial distribution.

Birkner et al. obtained necessary and sufficient conditions for the frequency between two independent and identically distributed continuous-state branching processes time-changed by a functional of the total mass process to be a Markov process. Foucart et al. extended this result to continuous-state branching processes with immigration. We generalize these results by dropping the independent and identically distributed assumption. Our result clarifies under which conditions a multi-type Λ-coalescent can be constructed from a multi-type branching process by a time change using the total mass. Finally, we address a problem formulated by Griffiths, by clarifying the relation between 2-type α-stable continuous-state branching processes and 2-type β-Fleming–Viot processes with mutation and selection.

We consider polynomial transforms (polyspectra) of Berry’s model – the Euclidean Random Wave model – and of Random Hyperspherical Harmonics. We determine the asymptotic behavior of variance for polyspectra of any order in the high-frequency limit. In particular, we are able to treat polyspectra of any odd order $q\ge 5$, whose asymptotic behavior was left as a conjecture in the case of Random Hyperspherical Harmonics by Marinucci and Wigman (Comm. Math. Phys. 2014). To this end, we exploit a relation between the variance of polyspectra and the distribution of uniform random walks on Euclidean space with finitely many steps, which allows us to rely on technical results in the latter context.

We calculate the large deviations for the length of the longest alternating subsequence and for the length of the longest increasing subsequence in a uniformly random permutation that avoids a pattern of length three. We treat all six patterns in the case of alternating subsequences. In the case of increasing subsequences, we treat two of the three patterns for which a classical large deviations result is possible. The same rate function appears in all six cases for alternating subsequences. This rate function is in fact the rate function for the large deviations of the sum of IID symmetric Bernoulli random variables. The same rate function appears in the two cases we treat for increasing subsequences. This rate function is twice the rate function for alternating subsequences.

We consider a discrete-time model for random interface growth which converges to the Polynuclear growth model in a particular limit. The height of the interface is initially flat and the evolution involves the addition of islands of height one according to a Poisson point process of nucleation events. The boundaries of these islands then spread in a stochastic manner, rather than at deterministic speed as in the Polynuclear growth model. The one-point distribution and multi-time distributions agree with point-to-line last passage percolation times in a geometric environment. An alternative interpretation for the growth model can be given through interacting particle systems experiencing pushing and blocking interactions.

Let ${\mathrm{\Lambda }}^{\ast }$ be the rate function in the large deviation principle for the sums $X_1+\cdots +X_n$ of independent identically distributed random variables $X_1,X_2,\dots$. It is shown that ${\mathrm{\Lambda }}^{\ast }(x)\sim -ln\mathsf{P}(X_1\ge x)$ (as $x\to \mathrm{\infty }$) if and only if $ln\mathsf{P}(X_1\ge x)\sim L_0(x)$ for some concave function $L_0$. The main ingredient of the proof is the general, explicit expression of a suitable quasi-minimizer in $t\ge 0$ of the Bernstein–Chernoff upper bound $e^{-tx}\mathsf{E}{e}^{t{X}_1}$ on $\mathsf{P}(X_1\ge x)$, which is amenable to analysis and, at the same time, is close enough to a true minimizer.

In this note, we establish a qualitative total variation version of Breuer–Major Central Limit Theorem for a sequence of the type $\frac{1}{\sqrt{n}}{\sum }_{1\le k\le n}f(X_k)$, where ${(X_k)}_{k\ge 1}$ is a centered stationary Gaussian process, under the hypothesis that the function f has Hermite rank $d\ge 1$ and belongs to the Malliavin space ${\mathbb{D}}^{1,2}$. This result in particular extends the recent works of [NNP21], where a quantitative version of this result was obtained under the assumption that the function f has Hermite rank $d=2$ and belongs to the Malliavin space ${\mathbb{D}}^{1,4}$. We thus weaken the ${\mathbb{D}}^{1,4}$ integrability assumption to ${\mathbb{D}}^{1,2}$ and remove the restriction on the Hermite rank of the base function. While our method is still based on Malliavin calculus, we exploit a particular instance of Malliavin gradient called the sharp operator, which reduces the desired convergence in total variation to the convergence in distribution of a bidimensional Breuer–Major type sequence.

The spine of the two-particle Fleming-Viot process driven by Brownian motion is not a Bessel-3 process.

We aim to set forth an extension of the result found in paper [6], which finds an explicit realisation of a reflecting Brownian motion with drift $-\mathrm{\mu }$, started at x, reflecting above zero, and its local time at zero. In this paper we find a corresponding realisation for a reflecting Brownian motion with drift $-\mathrm{\mu }$, started at x, reflected both above zero and below one, along with a corresponding expression in terms of associated local times, namely as the difference between the local time at zero and the local time at one.

We establish a central limit theorem for the eigenvalue counting function of a matrix of real Gaussian random variables.

Let $(\mathrm{\Omega },\mathcal{F})$ be a measurable space and $E\subset \mathrm{\Omega }\times \mathrm{\Omega }$. Suppose that $E\in \mathcal{F}\otimes \mathcal{F}$ and the relation on Ω defined as $x\sim y$⇔ $(x,y)\in E$ is reflexive, symmetric and transitive. Following [7], say that E is strongly dualizable if there is a sub-σ-field $\mathcal{G}\subset \mathcal{F}$ such that

$$\underset{P\in \mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })}{min}(1-P(E))=\underset{A\in \mathcal{G}}{max}|\mathrm{\mu }(A)-\mathrm{\nu }(A)|$$

for all probabilities μ and ν on $\mathcal{F}$. This paper investigates strong duality. Essentially, it is shown that E is strongly dualizable provided some mild modifications are admitted. Let ${\mathcal{G}}_0$ be the E-invariant sub-σ-field of $\mathcal{F}$. One result is that, for all probabilities μ and ν on $\mathcal{F}$, there is a probability ${\mathrm{\nu }}_0$ on $\mathcal{F}$ such that

$${\mathrm{\nu }}_0=\mathrm{\nu }\text{on}{\mathcal{G}}_0\text{and}\underset{P\in \mathrm{\Gamma }(\mathrm{\mu },{\mathrm{\nu }}_0)}{min}(1-P(E))=\underset{A\in {\mathcal{G}}_0}{max}|\mathrm{\mu }(A)-\mathrm{\nu }(A)|.$$

In the other results, $(\mathrm{\Omega },\mathcal{F})$ is a standard Borel space and the min over $\mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })$ is replaced by the inf over $\mathrm{\Gamma }(\mathrm{\mu },\mathrm{\nu })$ in the definition of strong duality. Then, E is strongly dualizable provided $\mathcal{G}$ is allowed to depend on $(\mathrm{\mu },\mathrm{\nu })$ or it is taken to be the universally measurable version of the E-invariant σ-field.

We consider the Sherrington-Kirkpatrick model with no external field and inverse temperature $\mathit{\beta }<1$ and prove that the expected operator norm of the covariance matrix of the Gibbs measure is bounded by a constant depending only on β. This answers an open question raised by Talagrand, who proved a bound of $C(\mathit{\beta }){(logn)}^8$. Our result follows by establishing an approximate formula for the covariance matrix which we obtain by differentiating the TAP equations and then optimally controlling the associated error terms. We complement this result by showing diverging lower bounds on the operator norm, both at the critical and low temperatures.

In this paper, we obtain the exact asymptotic behavior of Green functions of homogeneous random walks in ${\mathbb{Z}}^d$ killed at the first exit from and open cone of ${\mathbb{R}}^d$. Our approach combines methods of functional equations, integral representations of the Green function and Woess’ approach for the case of homogeneous random walks in ${\mathbb{Z}}^d$.

We consider a simplified model of first-passage percolation, involving two families of i.i.d. random variables $\{{\mathrm{\xi }}_{ij}\}$ and $\{{\mathrm{\eta }}_{ij}\}$ corresponding to the weights of the horizontal and vertical edges respectively. We obtain an explicit formula for the limiting shape of the first-passage distance expressed in terms of the corresponding limit shapes of the two sets of weights for the Seppäläinen–Johansson model. We also study the limiting fluctuations of this model when at least one of the sets of weights is Bernoulli distributed.

We prove that every locally finite vertex-transitive graph G admits a non-constant Lipschitz harmonic function.

In this note, we give a new and short proof for a theorem of Bodineau stating that the slab percolation threshold ${\hat{p}}_c$ for the FK-Ising model coincides with the standard percolation critical point $p_c$ in all dimensions $d\ge 3$. Both proofs rely on the positivity of the surface tension for $p>p_c$ proved by Lebowitz & Pfister. The key difference is that while Bodineau’s proof is based on a delicate dynamic renormalization inspired by the work of Barsky, Grimmett & Newman, our proof utilizes a technique of Benjamini & Tassion to prove the uniqueness of macroscopic clusters via sprinkling, which then implies percolation on slabs through a rather straightforward static renormalization.

Consider all the possible ways of coupling together two Brownian motions with the same starting position but with different drifts onto the same probability space. It is known that there exist couplings which make these processes agree for some random, positive, maximal initial length of time. Presently, we provide an explicit, elementary construction of such couplings.

Random matrices from the elliptic Ginibre orthogonal ensemble (GinOE) are a certain linear combination of a real symmetric, and real anti-symmetric, Gaussian random matrix and controlled by a parameter τ. Our interest is in the fluctuations of the number of real eigenvalues, for fixed τ when the expected number is proportional to the square root of the matrix size N, and for τ scaled to the weakly non-symmetric limit, when the expected number is proportional to N. By establishing that the generating function for the probabilities specifying the distribution of the number of real eigenvalues has only negative real zeros, and using too the fact that variances in both circumstances of interest tends to infinity as $N\to \mathrm{\infty }$, the known central limit theorem for the fluctuations is strengthened to a local central limit theorem, and the rate of convergence is discussed.

We consider a stochastic version of the point vortex system, in which the fluid velocity advects single vortices intermittently for small random times. Such system converges to the deterministic point vortex dynamics as the rate at which single components of the vector field are randomly switched diverges, and therefore it provides an alternative discretization of 2D Euler equations. The random vortex system we introduce preserves microcanonical statistical ensembles of the point vortex system, hence constituting a simpler alternative to the latter in the statistical mechanics approach to 2D turbulence.

In this note, we complete the analysis of the Martingale Wasserstein Inequality started in [5] by checking that this inequality fails in dimension $d\ge 2$ when the integrability parameter ρ belongs to $[1,2)$ while a stronger Maximal Martingale Wasserstein Inequality holds whatever the dimension d when $\mathit{\rho }\ge 2$.

Let $Z_1,Z_2,\dots$ be independent and identically distributed complex random variables with common distribution μ and set

$$P_n(z):=(z-Z_1)\cdots (z-Z_n).$$

Recently, Angst, Malicet and Poly proved that the critical points of $P_n$ converge in an almost-sure sense to the measure μ as n tends to infinity, thereby confirming a conjecture of Cheung-Ng-Yam and Kabluchko. In this short note, we prove for any fixed $k\in \mathbb{N}$, the empirical measure of zeros of the kth derivative of $P_n$ converges to μ in the almost sure sense, as conjectured by Angst-Malicet-Poly.

Consider an arbitrary large population at the present time, originated at an unspecified arbitrary large time in the past, where individuals within the same generation independently reproduce forward in time, sharing a common offspring distribution that may vary across generations. In other words, the reproduction is driven by a Galton-Watson process in a varying environment. The genealogy of the current generation, traced backward in time, is uniquely determined by the coalescent point process $(A_i,i\ge 1)$, where $A_i$ denotes the coalescent time between individuals i and $i+1$. In general, this process lacks the Markov property. In constant environment, Lambert and Popovic (2013) proposed a Markov process of point measures to reconstruct the coalescent point process. We provide a counterexample showing that their process lacks the Markov property. The main contribution of this work is to propose a vector valued Markov process $(B_i,i\ge 1)$, that can reconstruct the genealogy, with finite information for every i. Additionally, in the case of linear fractional offspring distributions, we establish that the variables of the coalescent point process $(A_i,i\ge 1)$ are independent and identically distributed.

HWI inequalities are interpolation inequalities relating entropy, Fisher information and optimal transport distances. We adapt an argument of Y. Wu for proving the Gaussian HWI inequality via a coupling argument to the discrete setting, establishing new interpolation inequalities for the discrete hypercube and the discrete torus. In particular, we obtain an improvement of the modified logarithmic Sobolev inequality for the discrete hypercube of Bobkov and Tetali.

We develop a generalisation of Mercer’s theorem to operator-valued kernels in infinite dimensional Hilbert spaces. We then apply our result to prove a Karhunen-Loève theorem, valid for mean-square continuous random functions valued in a separable Hilbert space. That is, we establish an orthogonal series expansion with uncorrelated coefficients for second-order random flows in a Hilbert space, that holds in mean-square uniformly over time.

We prove that the rescaled one-point fluctuations of the boundary of the percolation cluster in the Bernoulli-Exponential first passage percolation around the diagonal converge to a new family of distributions. The limit law is indexed by the rescaled level of percolation $s\ge 0$, it is Gaussian for $s=0$ and it converges to the Tracy–Widom distribution as $s\to \mathrm{\infty }$. For a fixed level $s>0$ the width of the cluster in the limit as a function of a time parameter t is of order $t^{2∕3}$ with Tracy–Widom fluctuations as in the discrete model.

We establish a Berry-Esseen theorem for random walks conditioned to stay positive under ${\mathbb{P}}^{+}$ (the probability by Doob’s h-transform), which quantifies the convergence rate in the Kolmogorov distance of the central limit theorem proved by Bryn-Jones and Doney (2006). Our approach is based on a recent analogous result by Grama and Xiao (2021) for random walks conditioned to stay positive over a finite time interval.

In H-percolation, we start with an Erdős–Rényi graph ${\mathcal{G}}_{n,p}$ and then iteratively add edges that complete copies of H. The process percolates if all edges missing from ${\mathcal{G}}_{n,p}$ are eventually added. We find the critical threshold $p_c$ when $H={\mathcal{G}}_{k,1∕2}$ is uniformly random, solving a problem of Balogh, Bollobás and Morris. In this sense, we find $p_c$ for most graphs H.

We study backward stochastic differential equations (BSDEs for short) on a probability space equipped with a Brownian filtration. We assume that the data are merely integrable. Moreover, the generator is imposed to satisfy monotonicity condition with respect to the value variable (with no restrictions on the growth) and to be Lipschitz continuous with respect to the control variable. We provide an a priori estimate and stability result for solutions to the studied BSDEs.

In the Potts spin glass model, inspired by the symmetry argument in [4] for the constrained free energy, we study the free energy with self-overlap correction. Similarly, we simplify the Parisi-type formula, originally an infimum over matrix-valued paths, into an optimization over real-valued paths, which can be interpreted as quantile functions of probability measures on the unit interval. Minimizers are usually called Parisi measures. Using results in [9] generalizing the Auffinger–Chen convexity argument, we deduce the uniqueness of the Parisi measure.

The approach in [4] is to perturb the covariance function of the Hamiltonian into one associated with the generic model, prove the results there, and then reduce the perturbation. Here, we choose to perturb the model by adding an external field parametrized by Ruelle probability cascades. This is used in the Hamilton–Jacobi equation approach and we directly apply results in [12].

Consider a random walk on ${\mathbb{Z}}^d$ in a translation-invariant and ergodic random environment and starting from the origin. In this short note, assuming that a quenched invariance principle for the opportunely-rescaled walks holds, we show how to derive an $L^1$-convergence of the corresponding semigroups. We then apply this result to obtain a quenched pathwise hydrodynamic limit for the simple symmetric exclusion process on ${\mathbb{Z}}^d$, $d\ge 2$, with i.i.d. symmetric nearest-neighbors conductances ${\mathit{\omega }}_{xy}\in [0,\mathrm{\infty })$ only satisfying

$$\mathbb{Q}({\mathit{\omega }}_{xy}>0)>p_c,$$

where $p_c$ is the critical value for bond percolation.

We observe that non-doubling metric spaces can be characterized as those that contain arbitrarily large sets of approximately equidistant points and use this to show that, for $\mathit{\gamma }\in (0,2]$, the γ-Liouville quantum gravity metric is almost surely not doubling and thus cannot be quasisymmetrically embedded into any finite-dimensional Euclidean space. This generalizes the corresponding result of Troscheit [34] for the Brownian map (which is equivalent to the case $\mathit{\gamma }=\sqrt{8∕3}$).

We show that the union of two or more independent uniform spanning forests (USF) on ${\mathbb{Z}}^d$ with $d\ge 3$ almost surely forms a connected transient graph. In fact, this also holds when taking the union of a deterministic everywhere percolating set and an independent ε-Bernoulli percolation on a single USF sample.

We consider a version of the classical Ehrenfest urn model with two urns and two types of balls: regular and heavy. Each ball is selected independently according to a Poisson process having rate 1 for regular balls and rate $\mathit{\alpha }\in (0,1)$ for heavy balls, and once a ball is selected, is placed in a urn uniformly at random. We study the asymptotic behavior when the total number of balls, N, goes to infinity, and the number of heavy ball is set to $m_N\in \{1,\dots ,N-1\}$. We focus on the observable given by the total number of balls in the left urn, which converges to a binomial distribution of parameter $1∕2$, regardless of the choice of the two parameters, α and $m_N$. We study the speed of convergence and show that this can exhibit three different phenomenologies depending on the choice of the two parameters of the model.

We consider fractal Boolean percolation on ${\mathbb{R}}^d$ with radius distribution a Radon measure ν on $(0,+\mathrm{\infty })$, $d\ge 2$ such that $\mathrm{\nu }{|}_{[1,+\mathrm{\infty })}$ has a finite d-moment. In this level of generality, the subcritical phase is not known to exist. We prove that under some regularity assumption on ν, the subcritical phase of the model, if existing, is sharp. Moreover, we prove that the existence of an unbounded connected component depends only on the fractal part (and not of the balls with radius larger than 1).

Place an active particle at the root of the infinite d-ary tree and dormant particles at each non-root site. Active particles move towards the root with probability p and otherwise move to a uniformly sampled child vertex. When an active particle moves to a site containing dormant particles, all the particles at the site become active. The critical drift $p_d$ is the infimum over all p for which infinitely many particles visit the root almost surely. We give improved bounds on ${sup}_{d\ge m}{p}_d$ and prove monotonicity of critical values associated to a self-similar variant.

Let ${\{X_t\}}_{t\ge 0}$ be a d-dimensional supercritical super-Brownian motion started from the origin with branching mechanism ψ. Denote by $R_t:=inf\{r>0:X_s(\{x\in {\mathbb{R}}^d:|x|\ge r\})=0,\forall 0\le s\le t\}$ the radius of the minimal ball (centered at the origin) containing the range of ${\{X_s\}}_{s\ge 0}$ up to time t. In [8], Pinsky proved that condition on non-extinction, ${lim}_{t\to \mathrm{\infty }}{R}_t∕t=\sqrt{2\mathit{\beta }}$ in probability, where $\mathit{\beta }:=-{\mathit{\psi }}^{\prime }(0)$. Afterwards, Engländer [1] studied the lower deviation probabilities of $R_t$. For the upper deviation probabilities, Engländer [1, Conjecture 8] conjectured that for $\mathit{\rho }>\sqrt{2\mathit{\beta }}$,

$$\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}log\mathbb{P}(R_t\ge \mathit{\rho }t|X_s({\mathbb{R}}^d)>0,\forall s>0)=-\left(\frac{{\mathit{\rho }}^2}{2}-\mathit{\beta }\right).$$

In this note, we confirmed this conjecture.

The paper contains the extension of Freedman’s inequality to the context of self-adjoint martingales on general finite von Neumann algebras.

We prove the existence of a finitely dependent proper colouring of the integer lattice ${\mathbb{Z}}^d$ that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for $d=1$, while only translation-invariant examples were known for higher d. Moreover we show that four colours suffice, and that the colouring can be expressed as an isometry-equivariant finitary factor of an i.i.d. process, with exponential tail decay on the coding radius. Our construction starts from known translation-invariant colourings and applies a symmetrization technique of possible broader utility.