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.

In the Deffuant model, individuals are located on the vertices of a graph, and are characterized by their opinion, a number in $[-1,1]$. The dynamics depends on two parameters: a confidence threshold $\mathit{\theta }<2$ and a convergent parameter ${\mathrm{\mu }}_{-}\le 1∕2$. Neighbors on the graph interact at rate one, which results in no changes if the neighbors disagree by more than θ, and a compromise with the opinions moving toward each other by a factor ${\mathrm{\mu }}_{-}$ if they disagree by less than θ (attraction). The main conjecture about the Deffuant model, which was proved for the process on the integers, states that, for all ${\mathrm{\mu }}_{-}>0$ and starting from the product measure in which the opinions are uniformly distributed in the interval $[-1,1]$, there is a phase transition from discordance to consensus at the confidence threshold one. In this paper, we study a natural variant of the model in which neighbors who disagree by more than θ feel more strongly about their own opinion, which is modeled by assuming that the opinions move away from each other by a divergent parameter ${\mathrm{\mu }}_{+}$ (repulsion). We prove, for the process on the integers, the absence of a phase transition even for arbitrarily small ${\mathrm{\mu }}_{+}>0$, in the sense that, for every nontrivial choice of θ, there is always discordance.

We provide sufficient criteria for explosion in Crump-Mode-Jagers branching process, via the process producing an infinite path in finite time. As an application, we deduce a phase-transition in the infinite tree associated with a class of recursive tree models with fitness, showing that in one regime every node in the tree has infinite degree, whilst in another, the tree is locally finite, with a unique infinite path. The latter class encompasses many models studied in the literature, including the weighted random recursive tree, the preferential attachment tree with additive fitness, and the Bianconi-Barabási model, or preferential attachment tree with multiplicative fitness.

Consider a stationary Poisson process of horospheres in a d-dimensional hyperbolic space. In the focus of this note is the total surface area these random horospheres induce in a sequence of balls of growing radius R. The main result is a quantitative, non-standard central limit theorem for these random variables as the radius R of the balls and the spatial dimension d tend to infinity simultaneously.

The aim of this paper is to establish Hoeffding and Bernstein type concentration inequalities for weighted sums of exchangeable random variables. A special case is the i.i.d. setting, where random variables are sampled independently from some distribution (and are therefore exchangeable). In contrast to the existing literature on this problem, our results provide a natural unified view of both the i.i.d. and the exchangeable setting.

We study a singular control problem for a Brownian motion with constant drift and variance reflected at the origin. Exerting control pushes the process towards the origin and generates a concave increasing state-dependent yield which is discounted at a fixed rate. We show that the solution to the problem is sometimes more complicated than anticipated. Indeed, for some parameter values, the optimal policy is a band policy with two reflecting barriers plus a repelling boundary where smooth fit fails. We also show that the apparent anomaly can be understood as involving a switch between two strategies with different risk profiles: The risk-neutral decision maker initially gambles on the more risky strategy and lowers risk if this strategy underperforms.

We present a lookdown construction for a Moran seed-bank model with variable active and inactive population sizes and we show that the empirical measure of our model coincides with that of the Seed-Bank-Moran Model with latency of Greven, den Hollander and Oomen [6]. Furthermore, we prove that the time to the most recent common ancestor, starting from N individuals with stationary distribution over its state (active or inactive), has the same asymptotic order as the largest inactivity period. Additionally, we give an explicit approximation of its distribution under extra assumtion on the inactivity and activity switching rates. We then find the first asymptotic order of the fixation time of a single beneficial mutant conditioned to invade the whole population, resulting to be of order $lnN$.

Strassen [23] established that there exists a two step martingale with marginal distributions μ, ν if and only if μ, ν are in convex order. Recently Choné, Gozlan and Kramarz [6] obtained a transport characterization of the stochastic order defined by convex positively 1-homogeneous functions, in the spirit of Strassen’s theorem under certain technical assumptions. In this note we prove the result of [6] in full generality. We also observe that the restriction of the result to the case where μ, ν are supported on a half space is equivalent to Strassen’s classical theorem.

Dynkin’s isomorphism Theorem as well as its derivatives, are identities in law providing the law of the sum of two independent processes. Most of the time these two processes are respectively the local time process of a Markov process and a permanental process. In spite of their various applications, these identities still remain mysterious in the sense that there is no natural reason for summing these two independent processes. We will present an explanation available in particular for the so-called “generalized second Ray-Knight Theorem”.

We study the short-time asymptotic behavior of the Brox diffusion. While the first order limit is a Brownian motion as in the standard diffusion cases, we show that the higher order terms have different convergence rates and limit processes that reflect the irregularity of the Brox diffusion.

We use local limits of Galton-Watson trees to establish local limit theorems for permutations conditioned to avoid a pattern of length three. In the case of $\mathbf{321}$-avoiding permutations our results resolve an open problem of Pinsky. In the other cases our results give new descriptions of the limiting objects in terms of size-biased Galton-Watson trees.

We prove that the number of points of a stationary linear Hawkes process lying in any bounded subset of the real line has exponential moments, without any other assumption than the one needed for existence of such stationary process, namely the spectral radius of the matrix of ${\mathbb{L}}^1$ norms of interaction functions is smaller than one. The proof relies on a mass transport principle argument. We also specify the dependence of the bounds with respect to the base rates and the matrix of ${\mathbb{L}}^1$ norms of interaction functions defining the Hawkes process and give a functional version of the result.

We prove the existence and regularity of the isochron map for stable invariant manifolds of a large class of evolution equations. Our results apply in particular to the isochron map of reaction-diffusion equations and neural field equations. Using the regularity properties proven here, we prove a strong Itô formula for the isochronal phase of stochastically perturbed travelling waves and other patterns appearing in SPDEs driven by white noise, even for SPDEs that only admit mild solutions.

We prove the local convergence of uniform bipartite maps with prescribed face degrees in the high genus regime. Unlike in the previous work [7] on the subject, we do not make any assumption on the tail of the face degrees, except that they remain finite in the limit.

Cyclic monotone independence is an algebraic notion of non-commutative independence, introduced in the study of multi-matrix random matrix models with small rank. Its algebraic form turns out to be surprisingly close to monotone independence, which is why it was named cyclic monotone independence. This paper conceptualizes this notion by showing that the same random matrix model is also a model for the monotone independence with an appropriately chosen state. This observation provides a unified nonrandom matrix model for both types of monotone independences.

In this note, we study the low temperature $(2+1)$ D SOS interface above a hard floor with critical pinning potential ${\mathit{\lambda }}_w=log(\frac{1}{1-e^{-4\mathit{\beta }}})$. At $\mathit{\lambda }<{\mathit{\lambda }}_w$ entropic repulsion causes the surface to delocalize and be rigid at height $\frac{1}{4\mathit{\beta }}logn+O(1)$; at $\mathit{\lambda }>{\mathit{\lambda }}_w$ it is localized at some $O(1)$ height. We show that at $\mathit{\lambda }={\mathit{\lambda }}_w$, there is delocalization, with rigidity now at height $\lfloor \frac{1}{6\mathit{\beta }}logn+\frac{1}{3}\rfloor$, confirming a conjecture of Lacoin.

Consider a log-correlated Gaussian field Γ and its associated imaginary multiplicative chaos $:e^{i\mathit{\beta }\mathrm{\Gamma }}$: where β is a real parameter. In [3], we showed that for any nonzero test function f, the law of $\int f:e^{i\mathit{\beta }\mathrm{\Gamma }}$: possesses a smooth density with respect to Lebesgue measure on $\mathbb{C}$. In this note, we show that this density is strictly positive everywhere on $\mathbb{C}$. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces.

In this note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form:

$$f(X_1+X_2)\stackrel{\text{d}}{=}f(X_1)+f(X_2),$$

where $X_1$ and $X_2$ represent two independent identically distributed real-valued random variables governed by a distribution μ having appropriate support. The symbol $\stackrel{d}{=}$ denotes equality in distribution. When μ is an exponential distribution, we provide sufficient (regularity) conditions on the function f to ensure that the unique measurable solution to the above equation is solely linear. Furthermore, we present some partial results in the general case, establishing a connection to integrated Cauchy functional equations.

We theoretically analyze the properties of a geodesic random walk on the Euclidean $d\text{-sphere}$. Specifically, we prove that the random walk’s transition kernel is Wasserstein contractive with a contraction rate which can be bounded from above independently of the dimension d. Our result is of particular interest due to its implications regarding the potential for dimension-independent performance of both geodesic slice sampling on the sphere and Gibbsian polar slice sampling, which are Markov chain Monte Carlo methods for approximate sampling from essentially arbitrary distributions on their respective state spaces.

We consider a spectrally negative branching Lévy process where the particles undergo dyadic branching and are killed when entering the negative half-line. The purpose of this short note is to give conditions under which this process dies out a.s., and then study the asymptotics of its all-time maximum.

We study the asymptotic behaviour of the number of times the elephant random walk in the critical regime visits the origin. Our result entails that most zeros of the critical elephant random walk occur shortly before its last passage time at zero. Additionally, we derive the tail estimate of the first return time of the random walk.

Given a general Itô semimartingale, its Markovian projection is an Itô process, with Markovian differential characteristics, that matches the one-dimensional marginal laws of the original process. We construct Markovian projections for Itô semimartingales with jumps, whose flows of one-dimensional marginal laws are solutions to non-local Fokker–Planck–Kolmogorov equations (FPKEs). As an application, we show how Markovian projections appear in building calibrated diffusion/jump models with both local and stochastic features.

We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges.

Complex solutions to squared Bessel SDEs appear naturally in relation to Schramm-Loewner evolutions. We prove the large deviation principle for such solutions as the dimension parameter tends to $-\mathrm{\infty }$.

We analyse the scaling of the weights added by invasion percolation on a uniformly weighted branching process tree. In this paper, we are interested in the invasion percolation cluster (IPC), obtained by performing invasion percolation for n steps and letting $n\to \mathrm{\infty }$. The volume scaling of the IPC was discussed in detail in [12] and in this work, we extend this analysis to the scaling of the cumulative weights.

We assume a power-law offspring distribution on the branching process tree with exponent α. For $\mathit{\alpha }>2$ and $\mathit{\alpha }\in (1,2)$, we observe a natural law-of-large-numbers result, where the cumulative weights scale similar to the volume, but converge to a different limit. However, for $\mathit{\alpha }<1$, where the weights added by invasion percolation vanish, the scaling changes significantly. For $\mathit{\alpha }\in (1∕2,1)$, the cumulative weights scale exponentially but with a different exponent than the volume scaling, while for $\mathit{\alpha }\in (0,1∕2)$ the cumulative weights are summable without any scaling. Such a phase transition at $\mathit{\alpha }=1∕2$ is novel and unexpected as there is no significant change in the neighbourhood scaling of the IPC around $\mathit{\alpha }=1∕2$.

We present a complete characterization of the asymptotic behaviour of a correlated Bernoulli sequence that depends on the parameter $\mathit{\theta }\in [0,1)$. A martingale theory based approach allows us to prove versions of the law of large numbers, quadratic strong law, law of iterated logarithm, almost sure central limit theorem and functional central limit theorem, in the case $\mathit{\theta }\le 1∕2$. For $\mathit{\theta }>1∕2$, we obtain a strong convergence to a non-degenerated random variable, including a central limit theorem and a law of iterated logarithm for the fluctuations.

Let $Q^d$ be the d-dimensional binary hypercube. We say that $P=\{v_1,\dots ,v_k\}$ is an increasing path of length $k-1$ in $Q^d$, if for every $i\in [k-1]$ the edge $v_i{v}_{i+1}$ is obtained by switching some zero coordinate in $v_i$ to a one coordinate in $v_{i+1}$.

Form a random subgraph $Q_p^d$ by retaining each edge in $E(Q^d)$ independently with probability p. We show that there is a phase transition with respect to the length of a longest increasing path around $p=\frac{e}{d}$. Let α be a constant and let $p=\frac{\mathit{\alpha }}{d}$. When $0<\mathit{\alpha }<e$, then there exists a $\mathit{\delta }\in (0,1)$ such that whp a longest increasing path in $Q_p^d$ is of length at most $(1-\mathit{\delta })d$. On the other hand, when $\mathit{\alpha }>e$, whp there is a path of length $d-2$ in $Q_p^d$, and in fact, whether it has length $d-2,d-1$, or d depends on whether the vertices $(0,\dots ,0)$ and $(1,\dots ,1)$ are in the giant connected component.

A Lévy processes resurrected in the positive half-line is a Markov process obtained by removing successively all jumps that make it negative. A natural question, given this construction, is whether the resulting process is absorbed at 0 or not. We first describe the law of the resurrected process in terms of that of the initial Lévy process. Then in many important classes of Lévy processes, we give conditions for absorption and conditions for non absorption bearing on the characteristics of the initial Lévy process.

We consider the Stochastic Heat Equation (SHE) in $(1+1)$ dimensions with delta Dirac initial data and spacetime white noise. We prove exact large-time asymptotics for multi-point correlations of the SHE for strictly sublinear space coordinates. The sublinear condition is optimal, in the sense that different asymptotics are known to occur when the space coordinates grow linearly [Lin23, Theorem 1.1]. A notable feature of our result is that the dependence on space coordinates of the SHE’s asymptotic multi-point correlations is given by the ground state of the delta-Bose gas.

We consider the number of roots of linear combinations of a system of n orthogonal eigenfunctions of a Sturm-Liouville initial value problem with i.i.d. standard Gaussian coefficients. We prove that its distribution inherits the asymptotic behavior of the number of roots of Quall’s random trigonometric polynomials. This result can be thought as a robustness result for the central limit theorem for the number of roots of Quall’s random trigonometric polynomials in the sense that small uniform perturbations of sines and cosines do not change the limit distribution.

The points of the closed range of a drift-free subordinator with no killing are used for separating into blocks the elements of a sample of size n from the standard exponential distribution. This gives rise to a random composition of n. Assuming that the subordinator has the Lévy measure, which behaves near zero like the gamma subordinator, we prove a law of the iterated logarithm for the number of blocks in the composition as n tends to infinity. Along the way we prove a law of the iterated logarithm for the Lebesgue convolution of a standard Brownian motion and a deterministic regularly varying function. This result may be of independent interest.

Inspired by recent advances in singular SPDE theory, we use the Poincaré inequality on Wiener space to show that controlled complementary Young regularity is sufficient to obtain Gaussian rough paths lifts. This allows us to completely bypass assumptions on the 2D variation regularity of the covariance and, as a consequence, we obtain cleaner proofs of approximation statements (with optimal convergence rates) and show the convergence of random Fourier series in rough paths metrics under minimal assumptions on the coefficients (which are sharper than those in the existent literature).

Dmitriev and Dynkin characterized the set of possible eigenvalues of Markov generators on a set of N points. We recover this result via decompositions in cycles. This probabilistic approach enables us to get more specific information about the possible eigenvalues when assumptions are made on the lengths of the cycles entering in the decomposition.

The decreasing Markov chain on {1,2,3, …} with transition probabilities $p(j,j-i)\propto 1∕i$ arises as a key component of the analysis of the beta-splitting random tree model. We give a direct and almost self-contained “probability” treatment of its occupation probabilities, as a counterpart to a more sophisticated but perhaps opaque derivation using a limit continuum tree structure and Mellin transforms.