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.