Schramm’s Locality Conjecture asserts that the value of the critical parameter $p_c$ of a graph satisfying $p_c<1$ depends only on its local structure. In this paper, we prove this conjecture in the particular case of transitive graphs with polynomial growth. Our proof relies on two recent works about such graphs, namely supercritical sharpness of percolation by the same authors and a finitary structure theorem by Tessera and Tointon.

Let $B={\{B_t\}}_{t\ge 0}$ be a one-dimensional standard Brownian motion and denote by $A_t,t\ge 0$, the quadratic variation of semimartingale $e^{B_t},t\ge 0$. The celebrated Bougerol’s identity in law (1983) asserts that, if $\mathit{\beta }={\{{\mathit{\beta }}_t\}}_{t\ge 0}$ is another Brownian motion independent of B, then ${\mathit{\beta }}_{A_t}$ has the same law as $sinh{B}_t$ for every fixed $t>0$. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving, as the second coordinates, the local times of B and β at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.

First, we prove the following sharp upper bound for the number of high degree differences in bipartite graphs. Let $(U,V,E)$ be a bipartite graph with $U=\{u_1,u_2,\dots ,u_n\}$ and $V=\{v_1,v_2,\dots ,v_n\}$. For $n\ge k>\frac{n}{2}$ we show that

$$\sum _{1\le i,j\le n}\mathbb{1}\{|\mathrm{deg}(u_i)-\mathrm{deg}(v_j)|\ge k\}\le 2k(n-k).$$

Second, as a corollary, we confirm the Burdzy–Pitman conjecture about the maximal spread of coherent and independent vectors: for $\mathit{\delta }\in (\frac{1}{2},1]$ we prove that

$$\mathbb{P}(|X-Y|\ge \mathit{\delta })\le 2\mathit{\delta }(1-\mathit{\delta })$$

for all random vectors $(X,Y)$ satisfying $X=\mathbb{P}(A|\mathcal{G})$ and $Y=\mathbb{P}(A|\mathcal{H})$ for some event A and independent σ-fields $\mathcal{G}$ and $\mathcal{H}$.

In this paper we consider stationary Markov chains with trivial two-sided tail sigma field, and prove that additive functionals satisfy the central limit theorem provided the variance of partial sums divided by n is bounded.

The Hierarchical Dirichlet process is a discrete random measure serving as an important prior in Bayesian non-parametrics. It is motivated with the study of groups of clustered data. Each group is modelled through a level two Dirichlet process and all groups share the same base distribution which itself is a drawn from a level one Dirichlet process. It has two concentration parameters with one at each level. The main results of the paper are the law of large numbers and large deviations for the hierarchical Dirichlet process and its mass when both concentration parameters converge to infinity. The large deviation rate functions are identified explicitly. The rate function for the hierarchical Dirichlet process consists of two terms corresponding to the relative entropies at each level. It is less than the rate function for the Dirichlet process, which reflects the fact that the number of clusters under the hierarchical Dirichlet process has a slower growth rate than under the Dirichlet process.

We investigate eigenvector statistics of the Truncated Unitary ensemble $\mathrm{TUE}(N,M)$ in the weakly non-unitary case $M=1$, that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when $M\ge N$, suggesting possible extensions to $\mathrm{TUE}(N,M)$ for all values of M.

In this short note, we prove that $v(-\mathit{\epsilon })=-v(\mathit{\epsilon })$. Here, $v(\mathit{\epsilon })$ is the speed of a one-dimensional random walk in a dynamic reversible random environment, that jumps to the right (resp. to the left) with probability $1∕2+\mathit{\epsilon }$ (resp. $1∕2-\mathit{\epsilon }$) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where $v(\mathit{\epsilon }),v(-\mathit{\epsilon })$ are well-defined, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only in the discrete setting.

In this note, we extend some recent results on systems of backward stochastic differential equations (BSDEs) with quadratic growth to the case of coupled forward-backward stochastic differential equations (FBSDEs). We work in a Markovian setting, and use results from the quadratic BSDE literature together with PDE techniques to obtain a-priori estimates which lead to an existence result. We also identify a general class of stochastic differential games whose corresponding FBSDE systems are covered by our main existence result. This leads to the existence of Markovian Nash equilibria for such games.

In this paper, we study almost sure central limit theorems for the spatial average of the solution to the stochastic wave equation in dimension $d\le 2$ over a Euclidean ball, as the radius of the ball diverges to infinity. This equation is driven by a general Gaussian multiplicative noise, which is temporally white and colored in space including the cases of the spatial covariance given by a fractional noise, a Riesz kernel, and an integrable function that satisfies Dalang’s condition.

We prove asymptotic formulas for the expectation of the vertex number and missed area of uniform random disc-polygons in convex disc-polygons. Our statements are the r-convex analogues of the classical results of Rényi and Sulanke [10] about random polygons in convex polygons.

We study a large class of McKean–Vlasov SDEs with drift and diffusion coefficient depending on the density of the solution’s time marginal laws in a Nemytskii-type of way. A McKean–Vlasov SDE of this kind arises from the study of the associated nonlinear FPKE, for which is known that there exists a bounded Sobolev-regular Schwartz-distributional solution u. Via the superposition principle, it is already known that there exists a weak solution to the McKean–Vlasov SDE with time marginal densities u. We show that there exists a strong solution the McKean–Vlasov SDE, which is unique among weak solutions with time marginal densities u. The main tool is a restricted Yamada–Watanabe theorem for SDEs, which is obtained by an observation in the proof of the classical Yamada–Watanabe theorem.

In [1] we considered periodic trees in which the number of children in successive generations is $(n,a_1,\dots ,a_k)$ with ${max}_i{a}_i\le C{n}^{1-\mathit{\delta }}$ and $(log{a}_i)∕logn\to b_i$ as $n\to \mathrm{\infty }$. Our proof contained an error. In this note we correct the proof. The theorem has changed: the critical value for local survival is asymptotically $\sqrt{{\overline{c}}_k(logn)∕n}$ where $l_k=max\{i:0\le i\le k,a_i\ne 1\}$ and ${\overline{c}}_k=min\{k+1-l_k-b_{l_k},(k-b)∕2\}$, where $b={lim}_{n\to \mathrm{\infty }}log(a_1{a}_2\cdots a_k)∕logn$.

We study the eigenvalue trajectories of a time dependent matrix $G_t=H+itv{v}^{\ast }$ for $t\ge 0$, where H is an $N\times N$ Hermitian random matrix and v is a unit vector. In particular, we establish that with high probability, an outlier can be distinguished at all times $t>1+N^{-1∕3+\mathit{\epsilon }}$, for any $\mathit{\epsilon }>0$. The study of this natural process combines elements of Hermitian and non-Hermitian analysis, and illustrates some aspects of the intrinsic instability of (even weakly) non-Hermitian matrices.

We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard’s iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingales. We work with a new outer measure which assigns zero value exactly to those properties which are instantly blockable.

A competition process on ${\mathbb{Z}}^d$ is considered, where two species compete to color the sites. The entities are driven by branching random walks. Specifically red (blue) particles reproduce in discrete time and place offspring according to a given reproduction law, which may be different for the two types. When a red (blue) particle is placed at a site that has not been occupied by any particle before, the site is colored red (blue) and keeps this color forever. The types interact in that, when a particle is placed at a site of opposite color, the particle adopts the color of the site with probability $p\in [0,1]$. Can a given type color infinitely many sites? Can both types color infinitely many sites simultaneously? Partial answers are given to these questions and many open problems are formulated.

The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continuity of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under φ-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Possible extensions of the results are discussed.

We present a very simple bijective proof of Cayley’s formula due to Foata and Fuchs (1970). This bijection turns out to be very useful when seen through a probabilistic lens; we explain some of the ways in which it can be used to derive probabilistic identities, bounds, and growth procedures for random trees with given degrees, including random d-ary trees. We also introduce a partial order on the degree sequences of rooted trees, and conjecture that it induces a stochastic partial order on heights of random rooted trees with given degrees.

Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found applications in various problems such as concentration inequalities and martingale optimal transport. In dimension one, it is well-known that the set of probability measures with a given mean is a lattice w.r.t. the convex order. Our main result is that, contrary to the minimum and maximum in the convex order, the Wasserstein projections are Lipschitz continuity w.r.t. the Wasserstein distance in dimension one. Moreover, we provide examples that show sharpness of the obtained bounds for the 1-Wasserstein distance.

We investigate the analytical properties of the α-Sun random variable, which arises from the domain of attraction of certain storage models involving a maximum and a sum. In the Fréchet case we show that this random variable is infinitely divisible, and we give the exact behaviour of the density at zero. In the Weibull case we give the exact behaviour of the density at infinity, and we show that the behaviour at zero is neither polynomial nor exponential. This answers the open questions in the recent paper [20].

We show that a one-dimensional regular continuous Markov process $\mathsf{X}$ with scale function $\mathfrak{s}$ is a Feller–Dynkin process precisely if the space transformed process $\mathfrak{s}(\mathsf{X})$ is a martingale when stopped at the boundaries of its state space. As a consequence, the Feller–Dynkin and the martingale property are equivalent for regular diffusions on natural scale with open state space. By means of a counterexample, we also show that this equivalence fails for multidimensional diffusions. Moreover, for Itô diffusions we discuss relations to Cauchy problems.

We prove a characterization of the Dirichlet–Ferguson measure over an arbitrary finite diffuse measure space. We provide an interpretation of this characterization in analogy with the Mecke identity for Poisson point processes.

In ballistic annihilation, infinitely many particles with randomly assigned velocities move across the real line and mutually annihilate upon contact. We introduce a variant with superimposed clusters of stationary particles, and provide a simple formula for the critical initial cluster density in terms of the mean and variance of the cluster size.

We study the limiting spectral distribution of large-dimensional sample covariance matrices associated with the order-d symmetric random tensors formed by products of d variables chosen from n independent standardized random variables. We find optimal sufficient conditions for this distribution to be the Marchenko-Pastur law in the case $d=d(n)$ and $n\to \mathrm{\infty }$. Our conditions reduce to $d^2=o(n)$ when the variables have uniformly bounded fourth moments. The proofs are based on a new concentration inequality for quadratic forms in symmetric random tensors and a law of large numbers for elementary symmetric random polynomials.

In this paper, it is shown that α-permanent in algebra is closely related to loop soup in probability. We give explicit expansions of α-permanents of the block matrices obtained from matrices associated to ∗-forests, which are a special class of matrices containing tridiagonal matrices. It is proved in two ways, one is the direct combinatorial proof, and the other is the probabilistic proof via loop soup.

Suppose that a random variable X of interest is observed. This paper concerns “the least favorable noise” ${\hat{Y}}_{\mathit{ϵ}}$, which maximizes the prediction error $E{[X-E[X|X+Y]]}^2$ (or minimizes the variance of $E[X|X+Y]$) in the class of Y with Y independent of X and $\mathrm{var}Y\le {\mathit{ϵ}}^2$. This problem was first studied by Ernst, Kagan, and Rogers ([4]). In the present manuscript, we show that the least favorable noise ${\hat{Y}}_{\mathit{ϵ}}$ must exist and that its variance must be ${\mathit{ϵ}}^2$. The proof of existence relies on a convergence result we develop for variances of conditional expectations. Further, we show that the function ${inf}_{\mathrm{var}Y\le {\mathit{ϵ}}^2}\mathrm{var}E[X|X+Y]$ is both strictly decreasing and right continuous in ϵ.

Let $(X_{n,j})$ and $(Y_{n,j})$ be two arrays of real random variables and $f:\mathbb{R}\to \mathbb{R}$ a Borel function. Define $D_n={\sum }_{j}f\left({\sum }_{i=1}^{j-1}{Y}_{n,i}\right)X_{n,j}$ and $D=Z\sqrt{{\int }_0^1{f}^2(B_{G(t)})dF(t)}$ where B is a standard Brownian motion, Z a standard normal random variable independent of B, and F and G are distribution functions. Conditions for $D_n\to D$, in distribution or stably, are given. Among other things, such conditions apply to certain sequences of stochastic integrals, when the quadratic variations of the integrand processes converge in distribution but not in probability. An upper bound for the Wasserstein distance between the probability distributions of $D_n$ and D is obtained as well.

We prove that the scaling limit of the weakly self-avoiding walk on a d-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is $o(V^{1∕2})$ where V is the volume (number of points) of the torus and if $d>4$. We also prove that the diffusion constant of the resulting torus Brownian motion is the same as the diffusion constant of the scaling limit of the usual weakly self-avoiding walk on ${\mathbb{Z}}^d$. This provides further manifestation of the fact that the weakly self-avoiding walk model on the torus does not feel that it is on the torus up until it reaches about $V^{1∕2}$ steps, which we believe is sharp.

Using the technique of evolving sets, we explore the connection between entropy growth and transience for simple random walks on connected infinite graphs with bounded degree. In particular we show that for a simple random walk starting at a vertex $x_0$, if the entropy after n steps, $E_n$ is at least $Cn$ where the C is independent of $x_0$, then the random walk is transient. We also give an example which demonstrates that the condition of C being independent of $x_0$ is necessary.

In modified two-neighbour bootstrap percolation in two dimensions each site of ${\mathbb{Z}}^2$ is initially independently infected with probability p and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time τ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected neighbours are required. More precisely, we show that for modified bootstrap percolation with high probability as $p\to 0$ it holds that

$$\mathit{\tau }\le exp\left(\frac{{\mathrm{\pi }}^2}{6p}-\frac{clog(1∕p)}{\sqrt{p}}\right)$$

for some positive constant c, while the classical model is known to lack the logarithmic factor.

In this paper we investigate the approximation of continuous functions on the Wasserstein space by smooth functions, with smoothness meant in the sense of Lions differentiability. In particular, in the case of a Lipschitz function we are able to construct a sequence of infinitely differentiable functions having the same Lipschitz constant as the original function. This solves an open problem raised in [12]. For (resp. twice) continuously differentiable function, we show that our approximation also holds for the first-order derivative (resp. second-order derivatives), therefore solving another open problem raised in [12].

The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in ${\mathbb{Z}}^d$, $d>2$, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from [2, 19] that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is ‘long’, i.e., the distance between its end-points is proportional to the diameter of the box. Our paper shows that, when $d>2$, such a critical threshold is strictly positive. In other words, the self-avoiding walk is long even in the presence of a positive density of monomers.

In this article, a new algorithm generating discrete uniform distribution on n elements from a binary random source is proposed by introducing the notion of coprimity and modulo function in number theory. The algorithm is a generalization of Von Neumann’s algorithm for $n=2$ [4] and Dijkstra’s algorithm for prime n [1]. The proof for the validity of the algorithm introduces the notion of random variable in residual classes, and combines the ideas and techniques from both probability theory and number theory.

In this work, we prove the well-posedness and propagation of chaos for a stochastic particle system in mean-field interaction under the assumption that the interacting kernel belongs to a suitable $L_t^q-L_x^p$ space. Contrary to the large deviation principle approach recently proposed in [2], the main ingredient of the proof here are the Partial Girsanov transformations introduced in [3] and developed in a general setting in this work.

We construct Dyson Brownian motion for $\mathit{\beta }\in (0,\mathrm{\infty }]$ by adapting the extrinsic construction of Brownian motion on Riemannian manifolds to the geometry of group orbits within the space of Hermitian matrices. When β is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits evolve by motion by (minus one half times) mean curvature.