We prove polynomial decay of the mixing field of the Vertex Reinforced Jump Process (VRJP) on ${\mathbb {Z}}^{2}$ with bounded conductances. Using [22] we deduce that the VRJP on ${\mathbb {Z}}^{2}$ with any constant conductances is almost surely recurrent. It gives a counterpart of the result of Merkl, Rolles [16] and Sabot, Zeng [22] for the 2-dimensional Edge Reinforced Random Walk.

This work is a continuation of [6], in which the same authors studied the fine structure of the extreme level sets of branching Brownian motion, namely the sets of particles whose height is within a finite distance from the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. Our main finding here is that most of the particles in an extreme level set come from only a small fraction of the clusters, which are atypically large.

The Kob-Andersen model is a fundamental example of a kinetically constrained lattice gas, that is, an interacting particle system with Kawasaki type dynamics and kinetic constraints. In this model, a particle is allowed to jump when sufficiently many neighboring sites are empty. We study the motion of a single tagged particle and in particular its convergence to a Brownian motion. Previous results showed that the path of this particle indeed converges in diffusive time-scale, and the purpose of this paper is to study the rate of decay of the self-diffusion coefficient for large densities. We find upper and lower bounds matching to leading behavior.

We study systems of simple point processes that admit stochastic intensities. We represent these point processes as thinnings of Poisson measures and are interested in a convergence result of such systems. This result states that, if the stochastic intensities of the limit point processes are independent of the underlying Poisson measures, the convergence in distribution in Skorohod topology of the stochastic intensities implies the same convergence for the point processes.

We consider two independent stationary random walks on large random regular graphs of degree $k\ge 3$ with N vertices. On these graphs, the exponential approximations of the meeting times are known to follow from existing methods and form a basis for the voter model’s diffusion approximations. The main result of this note improves the exponential approximations to an explicit form such that the first moments are asymptotically equivalent to $N(k-1)∕[2(k-2)]$.

Using Foster-Lyapunov techniques we establish new conditions on non-extinction, non-explosion, coming down from infinity and staying infinite, respectively, for the general continuous-state nonlinear branching processes introduced in Li et al. (2019). These results can be applied to identify boundary behaviors for the critical cases of the above nonlinear branching processes with power rate functions driven by Brownian motion and (or) stable Poisson random measure, which was left open in Li et al. (2019). In particular, we show that even in the critical cases, a phase transition happens between coming down from infinity and staying infinite.

In this paper, we study the small-time asymptotic behavior of the Kingman coalescent. We obtain the Berry-Esseen bound and the Edgeworth expansion in the central limit theorem. Moreover, by the method of mod-ϕ convergence, we also obtain the precise large deviations and the precise moderate deviations. Last, we also obtain a non-asymptotic deviation inequality for the Kingman coalescent.

We consider the endpoint large deviation for a continuous-time directed polymer in a Lévy-type random environment. When the space dimension is at least three, it is known that the so-called weak disorder phase exists, where the quenched and annealed free energies coincide. We prove that the rate function agrees with that of the underlying random walk near the origin in the whole interior of the weak disorder phase.

Infinite horizon optimal stopping problems for a Lévy processes with a two-sided reward function are considered. A two-sided verification theorem is presented in terms of the overall supremum and the overall infimum of the process. A result to compute the angle of the value function at the optimal thresholds of the stopping region is given. To illustrate the results, the optimal stopping problem of a compound Poisson process with two-sided exponential jumps and a two-sided payoff function is solved. In this example, the smooth-pasting condition does not hold.

This note is motivated by connections between the online and offline problems of selecting a possibly long subsequence from a Poisson-paced sequence of uniform marks under either a monotonicity or a sum constraint. The offline problem with the sum constraint amounts to counting the Poisson arrivals before their total exceeds a certain level. A precise asymptotics for the mean count is obtained by coupling with a nonlinear pure birth process.

Consider an infinite, connected, locally finite graph with vertex set $\mathcal{V}$. Intuitively a simple point process on $\mathcal{V}$ with attractive properties, should percolate more easily than a Bernoulli point process with the same marginales. Although it seems wrong to imagine that it could be true in general, we confirm this intuition on several examples involving Gaussian free fields and permanental free fields.

Generalizing the realized variance, the realized skewness (Neuberger, 2012) and the realized kurtosis (Bae and Lee, 2020), we construct realized cumulants with the so-called aggregation property. They are unbiased statistics of the cumulants of a martingale marginal based on sub-period increments of the martingale and its lower-order conditional cumulant processes. Our key finding is a relation between the aggregation property and the complete Bell polynomials. For an application we give an alternative proof and an extension of a cumulant recursion formula recently obtained by Lacoin et al. (2019) and Friz et al. (2020).

The sliced Wasserstein metric ${\mathcal{W̶}}_p$ and more recently max-sliced Wasserstein metric ${\bar{\mathcal{W}}}_p$ have attracted abundant attention in data sciences and machine learning due to their advantages to tackle the curse of dimensionality, see e.g. [15], [6]. A question of particular importance is the strong equivalence between these projected Wasserstein metrics and the (classical) Wasserstein metric ${\mathcal{W}}_p$. Recently, Paty and Cuturi have proved in [14] the strong equivalence of ${\bar{\mathcal{W}}}_2$ and ${\mathcal{W}}_2$. We show that the strong equivalence also holds for $p=1$, while the sliced Wasserstein metric does not share this nice property.

We consider the random hyperbolic graph model introduced by [KPK⁺10] and then formalized by [GPP12]. We show that, in the subcritical case $\alpha >1$, the size of the largest component is asymptotically almost surely $n^{1∕(2\alpha )+o(1)}$, thus strengthening a result of [BFM15] which gave only an upper bound of $n^{1∕\alpha +o(1)}$.

In this paper we consider the limiting distribution of KPZ growth models with random but not stationary initial conditions introduced in [6]. The one-point distribution of the limit is given in terms of a variational problem. By directly studying it, we deduce the right tail asymptotic of the distribution function. This gives a rigorous proof and extends the results obtained by Meerson and Schmidt in [18].

We consider branching random walk in random environment (BRWRE) and prove the existence of deterministic subsequences along which their maximum, centered at its mean, is tight. This partially answers an open question in [3]. The method of proof adapts an argument developed by Dekking and Host for branching random walks with bounded increments. The question of tightness without the need for subsequences remains open.

Ballisticity of a continuous self avoiding walk on hyperbolic spaces ${\mathbb{H}}^{\mathit{d}}$ is established.

We prove a dual Yamada–Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) Lévy processes. More precisely, we prove that weak uniqueness, i.e. uniqueness in law, implies weak joint uniqueness, i.e. joint uniqueness in law for the solution process and its driver.

This note provides an alternative probabilistic approach to the ${\mathrm{\Phi }}^{2\mathit{n}}$ theory in dimension 2. The key idea is to study the concentration phenomenon of martingales associated to polynomials of Gaussian variables. This is based on an adaptation of the work of Lacoin-Rhodes-Vargas [3] on the quantum Mabuchi K-energy.

We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian $d\times d$ matrix valued functions, with the same Poincaré constant. This generalizes recent results [ABY19, Kat20] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.

In a discrete-time financial market model with instantaneous price impact, we find an asymptotically optimal strategy for an investor maximizing her expected wealth. The asset price is assumed to follow a process with negative memory. We determine how the optimal growth rate depends on the impact parameter and on the covariance decay rate of the price.

For a family of random-cluster models with cluster weights $q\ge 1$, we prove that the probability that 0 is connected to x is asymptotically equal to $\frac{1}{q}\mathit{\chi }{(\mathit{\beta })}^2\mathit{\beta }{J}_{0,x}$ for $\mathit{\beta }<{\mathit{\beta }}_c$. The method developed in this article can be applied to any spin model for which there exists a random-cluster representation which is monotonic.

We show that the number of copies of a given rooted tree in a conditioned Galton–Watson tree satisfies a law of large numbers under a minimal moment condition on the offspring distribution.

We introduce the maximal correlation coefficient $R(M_1,M_2)$ between two noncommutative probability subspaces $M_1$and $M_2$ and show that the maximal correlation coefficient between the sub-algebras generated by $s_n:=x_1+\dots +x_n$and $s_m:=x_1+\dots +x_m$ equals $\sqrt{m∕n}$ for $m\le n$, where ${(x_i)}_{i\in \mathbb{N}}$ is a sequence of free and identically distributed noncommutative random variables. This is the free-probability analogue of a result by Dembo–Kagan–Shepp in classical probability. As an application, we use this estimate to provide another simple proof of the monotonicity of the free entropy and free Fisher information in the free central limit theorem. Moreover, we prove that the free Stein Discrepancy introduced by Fathi and Nelson is non-increasing along the free central limit theorem.

We consider the periodic Manhattan lattice with alternating orientations going north-south and east-west. Place obstructions on vertices independently with probability $0<p<1$. A particle is moving on the edges with unit speed following the orientation of the lattice and it will turn only when encountering an obstruction. The problem is that for which value of p is the trajectory of the particle closed almost surely. We prove this is true for $p>\frac{1}{2}-\mathit{\epsilon }$ with some $\mathit{\epsilon }>0$.

Let $1∕2\le \mathit{\beta }<1$, p be a generic prime number and $f_{\mathit{\beta }}$ be a random multiplicative function supported on the squarefree integers such that ${(f_{\mathit{\beta }}(p))}_p$ is an i.i.d. sequence of random variables with distribution $\mathbb{P}(f(p)=-1)=\mathit{\beta }=1-\mathbb{P}(f(p)=+1)$. Let $F_{\mathit{\beta }}$ be the Dirichlet series of $f_{\mathit{\beta }}$. We prove a formula involving measure-preserving transformations that relates the Riemann ζ function with the Dirichlet series of $F_{\mathit{\beta }}$, for certain values of β, and give an application. Further, we prove that the Riemann hypothesis is connected with the mean behavior of a certain weighted partial sum of $f_{\mathit{\beta }}$.

Consider a discrete time Markov chain with rather general state space which has an invariant probability measure μ. There are several sufficient conditions in the literature which guarantee convergence of all or μ-almost all transition probabilities to μ in the total variation (TV) metric: irreducibility plus aperiodicity, equivalence properties of transition probabilities, or coupling properties. In this work, we review and improve some of these criteria in such a way that they become necessary and sufficient for TV convergence of all respectively μ-almost all transition probabilities. In addition, we discuss so-called generalized couplings.

We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed Lévy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial Lévy process.

We introduce and analyze a generalization of the blocks spin Ising (Curie-Weiss) models that were discussed in a number of recent articles. In these block spin models each spin in one of s blocks can take one of a finite number of $q\ge 3$ values or colors, hence the name block spin Potts model. We prove a large deviation principle for the percentage of spins of a certain color in a certain block. These values are represented in an $s\times q$ matrix. We show that for uniform block sizes there is a phase transition. In some regime the only equilibrium is the uniform distribution of all colors in all blocks, while in other parameter regimes there is one predominant color, and this is the same color with the same frequency for all blocks. Finally, we establish log-Sobolev-type inequalities for the block spin Potts model.

We compute the average characteristic polynomial of the Hermitised product of M real or complex non-Hermitian Wigner matrices of size $N\times N$ with i.i.d. matrix elements, and the average of the characteristic polynomial of a product of M such matrices times the characteristic polynomial of the conjugate product matrix. Surprisingly, the results agree with that of the product of M real or complex Ginibre matrices at finite-N, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues. This extends the result of Forrester and Gamburd for one characteristic polynomial of such a single random matrix and only depends on the first two moments. In the limit $M\to \mathrm{\infty }$ at fixed N we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the Mth root. The position of the jth zero agrees asymptotically for large-j with the position of the jth Lyapunov exponent for products of Gaussian random matrices, hinting at the universality of the latter.

Is there a constant $r_0$ such that, in any invariant tree network linking rate-1 Poisson points in the plane, the mean within-network distance between points at Euclidean distance r is infinite for $r>r_0$? We prove a slightly weaker result. This is a continuum analog of a result of Benjamini et al (2001) on invariant spanning trees of the integer lattice.

We show that the local weak limit of a sequence of finite expander graphs with uniformly bounded degree is an ergodic (or extremal) unimodular random graph. In particular, the critical probability of percolation of the limiting random graph is constant almost surely. As a corollary, we obtain an improvement to a theorem by Benjamini-Nachmias-Peres (2011) in [4] on locality of percolation probability for finite expander graphs with uniformly bounded degree where we can drop the assumption that the limit is a single rooted graph.

We are concerned with the general problem of proving the existence of joint distributions of two discrete random variables M and N subject to infinitely many constraints of the form $\mathbb{P}(M=i,N=j)=0$. In particular, the variable M has a countably infinite range and the other variable N is uniformly distributed with finite range. The constraints placed on the joint distributions will require, for most elements j in the range of N, $\mathbb{P}(M=i,N=j)=0$ for infinitely many values of i in the range of M, where the corresponding values of i depend on j. To prove the existence of such joint distributions, we apply a theorem proved by Strassen on the existence of joint distributions with prespecified marginal distributions.

We consider some combinatorial structures that can be decomposed into components. Given $n\in \mathbb{N}$, consider an assembly, multiset, or selection $A_n$ among elements of $\{1,2,\dots ,n\}$, and consider a uniformly distributed random variable $N(n)$ on $A_n$. For each $i\le n$, denote by $C_i(n)$ the number of components of $N(n)$ of size i so that ${\sum }_{i\le n}i{C}_i(n)=n$. In each of these combinatorial structures, there exists infinitely many processes ${({(Z_i(n,x))}_{i\le n})}_x$, indexed by a real parameter x, consisting of non-negative independent variables ${(Z_i(n,x))}_{i\le n}$ such that the distribution of the vector ${(C_i(n))}_{i\le n}$ equals the distribution of the vector ${(Z_i(n,x))}_{i\le n}$ conditional on the event $\{{\sum }_{i\le n}i{Z}_i(n,x)=n\}$. Let $M(n,x)$ denote a random variable whose components are given by ${(Z_i(n,x))}_{i\le n}$. We introduce the notion of pivot mass which is then combined with Strassen’s work to provide couplings of $M(n,x)$ and $N(n)$ with desired properties. For each of these combinatorial structures, we prove that there exists a real number $x(n)$ for which we can couple $M(n,x)$ and $N(n)$ with ${\sum }_{i\le n}{(C_i(n)-Z_i(n,x))}^{+}\le 1$ when $x>x(n)$. We are providing a partial answer to the question “how much dependence is there in the process ${(C_i(n))}_{i\le n}$?”

Balls are sequentially allocated into n bins as follows: for each ball, an independent, uniformly random bin is generated. An overseer may then choose to either allocate the ball to this bin, or else the ball is allocated to a new independent uniformly random bin. The goal of the overseer is to reduce the load of the most heavily loaded bin after $\mathrm{\Theta }(n)$ balls have been allocated. We provide an asymptotically optimal strategy yielding a maximum load of $(1+o(1))\sqrt{\frac{8logn}{loglogn}}$ balls.

We study the long-time asymptotics of a network of weakly reinforced Pólya urns. In this system, which extends the WARM introduced by R. van der Hofstad et. al. (2016) to countable networks, the nodes fire at times given by a Poisson point process. When a node fires, one of the incident edges is selected with a probability proportional to its weight raised to a power $\mathit{\alpha }<1$, and then this weight is increased by 1.

We show that for $\mathit{\alpha }<1∕2$ on a network of bounded degrees, every edge is reinforced a positive proportion of time, and that the limiting proportion can be interpreted as an equilibrium in a countable network. Moreover, in the special case of regular graphs, this homogenization remains valid beyond the threshold $\mathit{\alpha }=1∕2$.

We show that the last zero before time t of a recurrent Bessel process with drift starting at 0 has the same distribution as the product of a right-censored exponential random variable and an independent beta random variable. This extends a recent result of Schulte-Geers and Stadje [19] from Brownian motion with drift to recurrent Bessel processes with drift. We give two proofs, one of which is intuitive, direct, and avoids heavy computations. For this we develop a novel additive decomposition for the square of a Bessel process with drift that may be of independent interest.

We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough path version of Donsker’s Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition.

Given any $\mathit{\delta }\in (0,\mathrm{\infty })$, let ${(X_n)}_{n=0}^{\mathrm{\infty }}$ be the δ-once reinforced random walk on ladder $\mathbb{Z}\times \{0,1\}$ with the following edge weight function at the $(n+1)$-th step:

$$w_n(e)=1+(\mathit{\delta }-1)\cdot I_{\{N(e,n)>0\}}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}N(e,n)=0,\hfill \\ \mathit{\delta }\hfill & \mathrm{if}N(e,n)>0.\hfill \end{array}\right.$$

Here $N(e,n):=\mathrm{\#}\{i<n:X_i{X}_{i+1}=e\}$ is the number of times that edge e has been traversed by the walk before time n. It was proved that ${(X_n)}_{n=0}^{\mathrm{\infty }}$ is almost surely recurrent for $\mathit{\delta }>1∕2$ (Vervoort (2002) [8] and Sellke (2006) [7]), while the a.s. recurrence for negative reinforcement factor $\mathit{\delta }\in (0,1∕2]$ remained open. In this note, we give an affirmative answer to this question.

We equip the edges of a deterministic graph H with independent but not necessarily identically distributed weights and study a generalized version of matchings (i.e. a set of vertex disjoint edges) in H satisfying the property that end-vertices of any two distinct edges are at least a minimum distance apart. We call such matchings as strong matchings and determine bounds on the expectation and variance of the minimum weight of a maximum strong matching. Next, we consider an inhomogenous random graph whose edge probabilities are not necessarily the same and determine bounds on the maximum size of a strong matching in terms of the averaged edge probability. We use local vertex neighbourhoods, the martingale difference method and iterative exploration techniques to obtain our desired estimates.

We use a renormalization of the total mass of the exit measure from the complement of a small ball centered at $x\in {\mathbb{R}}^d$ for $d\le 3$ to give a new construction of the total local time $L^x$ of super-Brownian motion at x.

We give a necessary and sufficient condition for strict convexity of the rate function of a random vector in ${\mathbb{R}}^d$. This condition is always satisfied when the random vector has finite Laplace transform. We also completely describe the effective domain of the rate function under a weaker condition.

In this paper we consider the cylindrical càdlàg property of a solution to a linear equation in a Hilbert space H, driven by a Levy process taking values in a possibly larger Hilbert space U. In particular, we are interested in diagonal type processes, where processes on coordinates are functionals of independent α-stable symmetric processes. We give the equivalent characterization in this case. We apply the same techniques to obtain a sufficient condition for existence of a càdlàg version of stable processes described as integrals of deterministic functions with respect to symmetric α-stable random measures with $\mathit{\alpha }\in [1,2)$.

We study a stochastic version of the classical Becker-Döring model, a well-known kinetic model for cluster formation that predicts the existence of a long-lived metastable state before a thermodynamically unfavorable nucleation occurs, leading to a phase transition phenomena. This continuous-time Markov chain model has received little attention, compared to its deterministic differential equations counterpart. We show that the stochastic formulation leads to a precise and quantitative description of stochastic nucleation events thanks to an exponentially ergodic quasi-stationary distribution for the process conditionally on nucleation has not yet occurred.

We prove that the best so far known constant $c_p=\frac{p^{-p}}{1-p},p\in (0,1)$ of a domination inequality, which originates to Lenglart, is sharp. In particular, we solve an open question posed by Revuz and Yor [12]. Motivated by the application to maximal inequalities, like e.g. the Burkholder-Davis-Gundy inequality, we also study the domination inequality under an additional monotonicity assumption. In this special case, a constant which stays bounded for p near 1 was proven by Pratelli and Lenglart. We provide the sharp constant for this case.

This paper concerns the parameter estimation problem for the quadratic potential energy in interacting particle systems from continuous-time and single-trajectory data. Even though such dynamical systems are high-dimensional, we show that the vanilla maximum likelihood estimator (without regularization) is able to estimate the interaction potential parameter with optimal rate of convergence simultaneously in mean-field limit and in long-time dynamics. This to some extend avoids the curse-of-dimensionality for estimating large dynamical systems under symmetry of the particle interaction.

To model the destruction of a resilient network, Cai, Holmgren, Devroye and Skerman introduced the k-cut model on a random tree, as an extension to the classic problem of cutting down random trees. Berzunza, Cai and Holmgren later proved that the total number of cuts in the k-cut model to isolate the root of a Galton–Watson tree with a finite-variance offspring law and conditioned to have n nodes, when divided by $n^{1-1∕2k}$, converges in distribution to some random variable defined on the Brownian CRT. We provide here a direct construction of the limit random variable, relying upon the Aldous–Pitman fragmentation process and a deterministic time change.

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that the random variable ${\mathit{\epsilon }}_1{v}_1+\cdots +{\mathit{\epsilon }}_n{v}_n$ lies in the Euclidean unit ball, where ${\mathit{\epsilon }}_1,\dots ,{\mathit{\epsilon }}_n\in \{-1,1\}$ are independent Rademacher random variables and $v_1,\dots ,v_n\in {\mathbb{R}}^d$ are fixed vectors of at least unit length. We extend some known results to the case that the ${\mathit{\epsilon }}_i$ are obtained from a Markov chain, including the general bounds first shown by Erdős in the scalar case and Kleitman in the vector case, and also under the restriction that the $v_i$ are distinct integers due to Sárközy and Szemeredi. In all extensions, the upper bound includes an extra factor depending on the spectral gap and additional dependency on the dimension. We also construct a pseudorandom generator for the Littlewood-Offord problem using similar techniques.

We study the spectral norm of random lifts of matrices. Given an $n\times n$ symmetric matrix A, and a centered distribution π on $k\times k(k\ge 2)$ symmetric matrices with spectral norm at most 1, let the matrix random lift $A^{(k,\mathrm{\pi })}$ be the random symmetric $kn\times kn$ matrix ${(A_{ij}{X}_{ij})}_{1\le i<j\le n}$, where $X_{ij}$ are independent samples from π. We prove that

$$\mathbb{E}\Vert A^{(k,\mathrm{\pi })}\Vert \lesssim \underset{i}{max}\sqrt{\sum _j{A}_{ij}^2}+\underset{ij}{max}|A_{ij}|\sqrt{log(kn)}.$$

This result can be viewed as an extension of existing spectral bounds on random matrices with independent entries, providing further instances where the multiplicative $\sqrt{logn}$ factor in the Non-Commutative Khintchine inequality can be removed.

As a direct application of our result, we prove an upper bound of $2(1+\mathit{ϵ})\sqrt{\mathrm{\Delta }}+O(\sqrt{log(kn)})$ on the new eigenvalues for random k-lifts of a fixed $G=(V,E)$ with $|V|=n$ and maximum degree Δ, compared to the previous result of $O(\sqrt{\mathrm{\Delta }log(kn)})$ by Oliveira [Oli10a] and the recent breakthrough by Bordenave and Collins [BC19] which gives $2\sqrt{\mathrm{\Delta }-1}+o(1)$ as $k\to \mathrm{\infty }$ for Δ-regular graph G.

We obtain upper Gaussian estimates of transition probabilities of a spatially non-homogeneous random walk confined to a multidimensional orthant. For the proof, we use comparison arguments based on discrete potential theory and variants of the Harnack principle.

We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices ${\mathbb{Z}}^d$ that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for the existence of an infinite cluster may be direction-dependent. Then, we prove that the phase transition in a given direction is sharp, and study the links between percolation and first-passage percolation on these oriented graphs.

We propose a new numerical method for one dimensional stochastic differential equations (SDEs). The main idea of this method is based on a representation of a weak solution of an SDE using a time-changed Brownian motion, which dates back to Doeblin (1940). In cases where the diffusion coefficient is bounded and is β-Hölder continuous with $0<\mathit{\beta }\le 1$, we provide the rate of strong convergence. An advantage of our approach is that we approximate the weak solution, which enables us to treat SDEs with no strong solution. Our scheme is the first to achieve strong convergence for the case of $0<\mathit{\beta }<1∕2$.

In this paper, we show several rigorous results on the phase transition of Finitary Random Interlacements (FRI). For the high intensity regime, we show the existence of a critical fiber length, and find its exact asymptotic as intensity goes to infinity. At the same time, our result for the low intensity regime proves the global existence of a non-trivial phase transition with respect to the system intensity.

We consider the symmetric exclusion process on the d-dimensional lattice with initial data invariant with respect to space shifts and ergodic. It is then known that as t diverges the distribution of the process at time t converges to a Bernoulli product measure. Assuming a summable decay of correlations of the initial data, we prove a quantitative version of this convergence by obtaining an explicit bound on the Ornstein $\overline{d}$-distance. The proof is based on the analysis of a two species exclusion process with annihilation.

We prove limit theorems for the weighted quadratic variation of trifractional Brownian motion and n-th order fractional Brownian motion. Furthermore, a sufficient condition for the $L^P$-convergence of the weighted quadratic variation for Gaussian processes is obtained as a byproduct. As an application, we give a statistical estimator for the self-similarity index of trifractional Brownian motion. These theorems extend results of Baxter, Gladyshev, and Norvaiša.

We consider the problem of inferring an ancestral state from observations at the leaves of a tree, assuming the state evolves along the tree according to a two-state symmetric Markov process. We establish a general branching rate condition under which maximum parsimony, a common reconstruction method requiring only the knowledge of the tree topology (but not of the substitution rates or other parameters), succeeds better than random guessing uniformly in the depth of the tree. We thereby generalize previous results of [13, 37]. Our results apply to both deterministic and i.i.d. edge weights.

We introduce the notion of localization at the boundary for conditioned random walks in i.i.d. and uniformly elliptic random environment on ${\mathbb{Z}}^d$, in dimensions two and higher. If $d=2$ or 3, we prove localization for (almost) all walks. In contrast, for $d\ge 4$, there is a phase transition for environments of the form ${\mathit{\omega }}_{\mathit{\epsilon }}(x,e)=\mathit{\alpha }(e)(1+\mathit{\epsilon }\mathrm{\xi }(x,e))$, where ${\{\mathrm{\xi }(x)\}}_{x\in {\mathbb{Z}}^d}$ is an i.i.d. sequence of random variables, and ε represents the amount of disorder with respect to a simple random walk.

We show fundamental properties of the Markov semigroup of recently proposed MCMC algorithms based on Piecewise-deterministic Markov processes (PDMPs) such as the Bouncy Particle Sampler, the Zig-Zag process or the Randomized Hamiltonian Monte Carlo method. Under assumptions typically satisfied in MCMC settings, we prove that PDMPs are Feller and that their generator admits the space of infinitely differentiable functions with compact support as a core. As we illustrate via martingale problems and a simplified proof of the invariance of target distributions, these results provide a fundamental tool for the rigorous analysis of these algorithms and corresponding stochastic processes.

An erratum for the paper is given.

We characterize all random point measures which are in a certain sense stable under the action of branching. Denoting by ⊛ the branching convolution operation introduced by Bertoin and Mallein, and by $\mathcal{Z}$ the law of a random point measure on the real line, we are interested in solutions to the fixed-point equation $\mathcal{E}=\mathcal{Z}⊛\mathcal{E}$, with $\mathcal{E}$ a random point measure distribution. Under suitable assumptions, we characterize all solutions of this equation as shifted decorated Poisson point processes with a uniquely defined shift.

The bisection method for kinetically constrained models (KCM) of Cancrini, Martinelli, Roberto and Toninelli is a vital technique applied also beyond KCM. In this note we present a new way of performing it, based on a novel two-block dynamics with a probabilistic proof instead of the original spectral one. We illustrate the method by very directly proving an upper bound on the relaxation time of KCM like the one for the East model in a strikingly general setting. Namely, we treat KCM on finite or infinite one-dimensional volumes, with any boundary condition, conditioned on any of the irreducible components of the state space, with arbitrary site-dependent state spaces and, most importantly, arbitrary inhomogeneous rules.

This paper obtains Hölder continuity of the mild solution to a stochastic parabolic equation defined on a bounded domain with homogeneous Dirichlet boundary condition. The Gaussian noise involved in the equation is white in time and correlated in space. The approach is based on a new Hölder-type estimate for the Green function in the domain.

We consider one infinite path of a Random Walk in Random Environment (RWRE, for short) in an unknown environment. This environment consists of either i.i.d. site or bond randomness. At each position the random walker stops and tells us the environment it sees at the point where it is, without telling us, where it is. These observations ${\mathit{\chi }}^{\prime }$ are spoiled by reading errors that occur with probability $p<1$. We show: If the RWRE is recurrent and satisfies the standard assumptions on such RWREs, then with probability one in the environment, the errors, and the random walk we are able reconstruct the law of the environment. For most situations this result is even independent of the value of p. If the distribution of the environment has a non-atomic part, we can even reconstruct the environment itself, up to translation.

We show that the multi–species higher spin stochastic vertex model, also called the $U_q(A_n^{(1)})$ vertex model, satisfies a duality where the indicator function has the form $\{{\mathrm{\eta }}_{[i,n]}^x\ge {\mathrm{\xi }}_{[i,n]}^x\}$. In other words, for every particle in the ξ configuration of species i at vertex x, there must be a particle of species $j\ge i$ at vertex x in the η configuration. For these duality functions, the dual process has fewer particles than the original process, making it suitable for applications. The proof follows by applying charge reversal to previously discovered duality functions, which also results in open boundary conditions. As a special case, we recover the duality for the stochastic six vertex model recently found by Y. Lin.

In [10], Jonasson and Steif conjectured that no non-degenerate sequence of transitive Boolean functions ${(f_n)}_{n⩾1}$ with ${lim}_{n\to \mathrm{\infty }}I(f_n)=\mathrm{\infty }$ could be tame (with respect to some ${(p_n)}_{n⩾1}$). In a companion paper [5], the author showed that this conjecture in its full generality is false, by providing a counter-example for the case when, at the same time, ${lim}_{n\to \mathrm{\infty }}n{p}_n=\mathrm{\infty }$ and ${lim}_{n\to \mathrm{\infty }}{n}^{\mathit{\alpha }}{p}_n=0$ for some $\mathit{\alpha }\in (0,1)$. In this paper we show that with slightly different assumptions, the conclusion of the conjecture holds when the sequence ${(p_n)}_{n⩾1}$ is bounded away from zero and one.

We study the phase transition of the free energy in the Sherrington-Kirkpatrick mean-field spin glass model with centered Gaussian external field. We show that the corresponding Almeida-Thouless line is the correct transition curve that distinguishes between the replica symmetric and replica symmetry breaking solutions in the Parisi formula.

We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a Poisson random number of edges are added between them. The process is parametrised by a positive real λ which governs the limiting average degree. We show that for each value of λ there is a unique random connected rooted multigraph $M(\mathit{\lambda })$ invariant under this evolution. As a consequence, starting from any finite graph G the process will almost surely converge in distribution to $M(\mathit{\lambda })$, which does not depend on G. We show that this limit has finite expected size. The same process naturally extends to one in which connectedness is not necessarily preserved, and we give a sharp threshold for connectedness of this version.

This is an asynchronous version, which is more realistic from the real-world network point of view, of a process we studied in [8, 9].

For random systems subject to a constraint, the microcanonical ensemble requires the constraint to be met by every realisation (‘hard constraint’), while the canonical ensemble requires the constraint to be met only on average (‘soft constraint’). It is known that for random graphs subject to topological constraints breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-vanishing specific relative entropy of the two ensembles. We investigate to what extent breaking of ensemble equivalence is manifested through the largest eigenvalue of the adjacency matrix of the graph. We consider two examples of constraints in the dense regime: (1) fix the degrees of the vertices (= the degree sequence); (2) fix the sum of the degrees of the vertices (= twice the number of edges). Example (1) imposes an extensive number of local constraints and is known to lead to breaking of ensemble equivalence. Example (2) imposes a single global constraint and is known to lead to ensemble equivalence. Our working hypothesis is that breaking of ensemble equivalence corresponds to a non-vanishing difference of the expected values of the largest eigenvalue under the two ensembles. We verify that, in the limit as the size of the graph tends to infinity, the difference between the expected values of the largest eigenvalue in the two ensembles does not vanish for (1) and vanishes for (2). A key tool in our analysis is a transfer method that uses relative entropy to determine whether probabilistic estimates can be carried over from the canonical ensemble to the microcanonical ensemble, and illustrates how breaking of ensemble equivalence may prevent this from being possible.

A finite form of de Finetti’s representation theorem is established using elementary information-theoretic tools: The distribution of the first k random variables in an exchangeable binary vector of length $n\ge k$ is close to a mixture of product distributions. Closeness is measured in terms of the relative entropy and an explicit bound is provided.

We prove a quenched invariance principle for continuous-time random walks in a dynamically averaging environment on $\mathbb{Z}$. In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.

We study coexistence in discrete time multi-type frog models. We first show that for two types of particles on ${\mathbb{Z}}^d$, for $d\ge 2$, for any jumping parameters $p_1,p_2\in (0,1]$, coexistence occurs with positive probability for sufficiently rich deterministic initial configuration. We extend this to the case of random distribution of initial particles. We study the question of coexistence for multiple types and show positive probability coexistence of $2^d$ types on ${\mathbb{Z}}^d$ for rich enough initial configuration. We also show an instance of infinite coexistence on ${\mathbb{Z}}^d$ for $d\ge 3$ provided we have sufficiently rich initial configuration.

The celebrated result by Biane-Bougerol-O’Connell relates Duistermaat-Heckman (DH) measures for coadjoint orbits of a compact Lie group G with the multi-dimensional Pitman transform of the Wiener process on its Cartan subalgebra. The DH theory admits several non-trivial generalizations. In this paper, we consider the case of $G=SU(2)$, and we give an interpretation of DH measures for $SU(2)\cong S^3$ valued moment maps in terms of an interesting stochastic process on the unit disc, and an interpretation of the DH measures for Poisson ${\mathbb{H}}^3$ valued moment maps (in the sense of Lu) in terms of a stochastic process on the interior of a hyperbola.

In this note, we give a new proof of Liggett’s theorem on the invariant measures of independent particle systems from [11] in the particular case of independent drifted Brownian motions. This particular case has received a lot of attention recently due to its applications for the analysis of the local extrema of discrete Gaussian free field. The novelty of our proof is that it identifies directly the expected Poisson Point Process with exponential intensity without relying on the Choquet-Deny convolution equation $\mathrm{\mu }\ast P=\mathrm{\mu }$ ([6, 7]).

The problem of the quickest detection of a change in the drift of a time-homogeneous diffusion process is considered under the assumption that the detection delay is exponentially penalized. In this framework, the past literature has shown that a two- or three-dimensional optimal stopping problem needs to be faced. In this note, we show how a change of measure significantly simplifies the setting by reducing the dimension of the optimal stopping problem to one or two, respectively. We illustrate this result in the well known Brownian motion case analyzed by Beibel [4] and when a Bessel process is observed, generalizing therefore the results for the linear penalty case obtained by Johnson and Peskir [13].