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.