This paper concerns the McKean–Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its role in connecting weak solutions to McKean–Vlasov SDEs with common noise and solutions to corresponding stochastic partial differential equations (SPDEs). By keeping track of the dependence structure between all components in a sequence of approximating processes, a compactness argument is employed to prove the existence of a weak solution assuming boundedness and joint continuity of the coefficients (allowing for degenerate diffusions). Weak uniqueness is established when the private (idiosyncratic) noise’s diffusion coefficient is nondegenerate and the drift is regular in the total variation distance. This seems sharp when one considers using finite-dimensional noise to regularise an infinite dimensional problem. The proof relies on a suitably tailored cost function in the Monge–Kantorovich problem and representation of weak solutions via Girsanov transformations.

We study the heat kernel and the Green’s function on the infinite supercritical percolation cluster in dimension $\mathit{d}\ge 2$ and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in (Electron. J. Probab. 14 (2009) 1–27). The proof relies on a structure of renormalization for the infinite percolation cluster introduced in (Comm. Pure Appl. Math. 71 (2018) 1717–1849), Gaussian bounds on the heat kernel established by Barlow in (Ann. Probab. 32 (2004) 3024–3084) and tools of the theory of quantitative stochastic homogenization. An important step in the proof is to establish a ${\mathit{C}}^{0,1}$-large-scale regularity theory for caloric functions on the infinite cluster and is of independent interest.

We are interested in the nearly supercritical regime in a family of max-type recursive models studied by Collet, Eckman, Glaser and Martin (Comm. Math. Phys. 94 (1984) 353–370) and by Derrida and Retaux (J. Stat. Phys. 156 (2014) 268–290) and prove that, under a suitable integrability assumption on the initial distribution, the free energy vanishes at the transition with an essential singularity with exponent $\frac{1}{2}$. This gives a weaker answer to a conjecture of Derrida and Retaux (J. Stat. Phys. 156 (2014) 268–290). Other behaviours are obtained when the integrability condition is not satisfied.

Sandpile dynamics are considered on graphs constructed from periodic plane and space tilings by assigning a growing piece of the tiling, either torus or open boundary conditions. A general method of obtaining the Green’s function of the tiling is given, and a total variation cut-off phenomenon is demonstrated under general conditions. It is shown that the boundary condition does not affect the mixing time for planar tilings. In a companion paper, computational methods are used to demonstrate that an open boundary condition alters the mixing time for the D4 lattice in dimension 4, while an asymptotic evaluation shows that it does not change the asymptotic mixing time for the cubic lattice ${\mathbb{Z}}^{\mathit{d}}$ for all sufficiently large d.

Consider the 3D Ising model on a box of side length n with minus boundary conditions above the $\mathit{x}\mathit{y}$-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height ${\mathit{M}}_{\mathit{n}}$ of this interface: for every β large, ${\mathit{M}}_{\mathit{n}}/log\mathit{n}\to {\mathit{c}}_{\mathit{\beta }}$ in probability as $\mathit{n}\to \infty$.

Here, we show that the laws of the centered maxima ${({\mathit{M}}_{\mathit{n}}-\mathbb{E}[{\mathit{M}}_{\mathit{n}}])}_{\mathit{n}\ge 1}$ are uniformly tight. Moreover, even though this sequence does not converge, we prove that it has uniform upper and lower Gumbel tails (exponential right tails and doubly exponential left tails). Key to the proof is a sharp (up to $\mathit{O}(1)$ precision) understanding of the surface large deviations. This includes, in particular, the shape of a pillar that reaches near-maximum height, even at its base, where the interactions with neighboring pillars are dominant.

We introduce diffusions on a space of interval partitions of the unit interval that are stationary with the Poisson–Dirichlet laws with parameters $\left(\mathit{\alpha },0\right)$ and $\left(\mathit{\alpha },\mathit{\alpha }\right)$. The construction has two steps. The first is a general construction of interval partition processes obtained previously by decorating the jumps of a Lévy process with independent excursions. Here, we focus on the second step which requires explicit transition kernels and, what we call, pseudo-stationarity. This allows us to study processes obtained from the original construction via scaling and time-change. In a sequel paper we establish connections to diffusions on decreasing sequences introduced by Ethier and Kurtz (Adv. in Appl. Probab. 13 (1981) 429–452) and Petrov (Funktsional. Anal. i Prilozhen. 43 (2009) 45–66). The latter diffusions are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction is also a step toward resolving longstanding conjectures by Feng and Sun on measure-valued Poisson–Dirichlet diffusions and by Aldous on a continuum-tree-valued diffusion.

We consider the narrow wedge solution to the Kardar–Parisi–Zhang stochastic PDE under the characteristic $3:2:1$ scaling of time, space and fluctuations. We study the correlation of fluctuations at two different times. We show that, when the times are close to each other, the correlation approaches one at a power-law rate with exponent $2/3$, while, when the two times are remote from each other, the correlation tends to zero at a power-law rate with exponent $-1/3$. We also prove exponential-type tail bounds for differences of the solution at two space-time points.

Three main tools are pivotal to proving these results: (1) a representation for the two-time distribution in terms of two independent narrow wedge solutions, (2) the Brownian Gibbs property of the KPZ line ensemble and (3) recently proved one-point tail bounds on the narrow wedge solution.

We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable as well as a stochastic version of the standard “twist condition” frequently used in deterministic Monge transport theory. This setting includes, in particular, the case of the distance cost $\mathit{c}(\mathit{x},\mathit{y})=|\mathit{x}-\mathit{y}|$. We prove existence and uniqueness of the solution and characterize it as the first time Brownian motion hits a barrier that is determined by solutions to a corresponding dual problem.

We study the scaling limit of a branching random walk in static random environment in dimension $\mathit{d}=1,2$ and show that it is given by a super-Brownian motion in a white noise potential. In dimension 1 we characterize the limit as the unique weak solution to the stochastic PDE

$${\partial }_{\mathit{t}}\mathit{\mu }=(\mathrm{\Delta }+\mathit{\xi })\mathit{\mu }+\sqrt{2\mathit{\nu }\mathit{\mu }}\tilde{\mathit{\xi }}$$

for independent space white noise ξ and space-time white noise $\tilde{\mathit{\xi }}$. In dimension 2 the study requires paracontrolled theory and the limit process is described via a martingale problem. In both dimensions we prove persistence of this rough version of the super-Brownian motion.

We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type

$$\mathit{d}{\mathit{X}}_{\mathit{t}}=\text{V}({\mathit{X}}_{\mathit{t}},\mathcal{L}({\mathit{X}}_{\mathit{t}}))\mathit{d}\mathit{t}+\text{F}({\mathit{X}}_{\mathit{t}},\mathcal{L}({\mathit{X}}_{\mathit{t}}))\mathit{d}{\mathit{W}}_{\mathit{t}},$$

where W is a random rough path and $\mathcal{L}({\mathit{X}}_{\mathit{t}})$ is the law of ${\mathit{X}}_{\mathit{t}}$. We prove propagation of chaos, and provide also an explicit optimal convergence rate. The analysis is based upon the tools we developed in our companion paper (Electron. J. Probab. 25 (2020) 21) for solving mean field rough differential equations and in particular upon a corresponding version of the Itô-Lyons continuity theorem. The rate of convergence is obtained by a coupling argument developed first by Sznitman for particle systems with Brownian inputs.

In this article, we study the smallest gaps of the circular β-ensemble (CβE) on the unit circle, where β is any positive integer. The main result is that the smallest gaps, after being normalized by ${\mathit{n}}^{\frac{\mathit{\beta }+2}{\mathit{\beta }+1}}$, will converge in distribution to a Poisson point process with some explicit intensity. And thus one can derive the limiting density of the kth smallest gap, which is proportional to ${\mathit{x}}^{\mathit{k}(\mathit{\beta }+1)-1}{\mathit{e}}^{-{\mathit{x}}^{\mathit{\beta }+1}}$. In particular, the results apply to the classical COE, CUE and CSE in random matrix theory. The essential part of the proof is to derive several identities and inequalities regarding the Selberg integral, which should have their own interest.

Linde, Moore and Nordahl introduced a generalisation of the honeycomb dimer model to higher dimensions. The purpose of this article is to describe a number of structural properties of this generalised model. First, it is shown that the samples of the model are in one-to-one correspondence with the perfect matchings of a hypergraph. This leads to a generalised Kasteleyn theory: the partition function of the model equals the Cayley hyperdeterminant of the adjacency hypermatrix of the hypergraph. Second, we prove an identity which relates the covariance matrix of the random height function directly to the random geometrical structure of the model. This identity is known in the planar case but is new for higher dimensions. It relies on a more explicit formulation of Sheffield’s cluster swap which is made possible by the structure of the honeycomb dimer model. Finally, we use the special properties of this explicit cluster swap to give a new and simplified proof of strict convexity of the surface tension in this case.