Consider a uniform rooted Cayley tree ${\mathit{T}}_{\mathit{n}}$ with n vertices and let m cars arrive sequentially, independently, and uniformly on its vertices. Each car tries to park on its arrival node, and if the spot is already occupied, it drives towards the root of the tree and parks as soon as possible. Lackner and Panholzer (J. Combin. Theory Ser. A 142 (2016) 1–28) established a phase transition for this process when $\mathit{m}\approx \frac{\mathit{n}}{2}$. In this work, we couple this model with a variant of the classical Erdős–Rényi random graph process. This enables us to describe the phase transition for the size of the components of parked cars using a modification of the multiplicative coalescent which we name the frozen multiplicative coalescent. The geometry of critical parked clusters is also studied. Those trees are very different from Bienaymé–Galton–Watson trees and should converge towards the growth-fragmentation trees canonically associated to the $3/2$-stable process that already appeared in the study of random planar maps.

We investigate the stationary measure π of SDEs driven by additive fractional noise with any Hurst parameter and establish that π admits a smooth Lebesgue density obeying both Gaussian-type lower and upper bounds. The proofs are based on a novel representation of the stationary density in terms of a Wiener–Liouville bridge, which proves to be of independent interest: We show that it also allows to obtain Gaussian bounds on the nonstationary density, which extend previously known results in the additive setting. In addition, we study a parameter-dependent version of the SDE and prove smoothness of the stationary density, jointly in the parameter and the spatial coordinate. With this, we revisit the fractional averaging principle of Li and Sieber (Ann. Appl. Probab. 32 (2022) 3964–4003) and remove an ad hoc assumption on the limiting coefficients. Avoiding any use of Malliavin calculus in our arguments, we can prove our results under minimal regularity requirements.

We study the Loewner evolution whose driving function is ${\mathit{W}}_{\mathit{t}}={\mathit{B}}_{\mathit{t}}^1\mathbf{+}\mathit{i}{\mathit{B}}_{\mathit{t}}^2$, where $({\mathit{B}}^1,{\mathit{B}}^2)$ is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm–Loewner evolution (SLE) with complex parameter values. We show that our Loewner evolutions behave very differently from ordinary SLE. For example, if neither ${\mathit{B}}^1$ nor ${\mathit{B}}^2$ is identically equal to zero, then the set of points disconnected from ∞ by the Loewner hull has nonempty interior at each time. We also show that our model exhibits three phases analogous to the phases of SLE: a phase where the hulls have zero Lebesgue measure, a phase where points are swallowed but not hit by the hulls and a phase where the hulls are space-filling. The phase boundaries are expressed in terms of the signs of explicit integrals. These boundaries have a simple closed form when the correlation of the two Brownian motions is zero.

We consider invasion percolation on the complete graph ${\mathit{K}}_{\mathit{n}}$, started from some number $\mathit{k}(\mathit{n})$ of distinct source vertices. The outcome of the process is a forest consisting of $\mathit{k}(\mathit{n})$ trees, each containing exactly one source. Let ${\mathit{M}}_{\mathit{n}}$ be the size of the largest tree in this forest. Logan, Molloy and Pralat (2018) proved that if $\mathit{k}(\mathit{n})/{\mathit{n}}^{1/3}\to 0$ then ${\mathit{M}}_{\mathit{n}}/\mathit{n}\to 1$ in probability. In this paper, we prove a complementary result: if $\mathit{k}(\mathit{n})/{\mathit{n}}^{1/3}\to \infty$, then ${\mathit{M}}_{\mathit{n}}/\mathit{n}\to 0$ in probability. This establishes the existence of a phase transition in the structure of the invasion percolation forest around $\mathit{k}(\mathit{n})\asymp {\mathit{n}}^{1/3}$.

Our arguments rely on the connection between invasion percolation and critical percolation, and on a coupling between multisource invasion percolation with differently-sized source sets. A substantial part of the proof is devoted to showing that, with high probability, a certain fragmentation process on large random binary trees leaves no components of macroscopic size.

A Poisson system is a Poisson point process and a group action, together forming a measure-preserving dynamical system. Ornstein and Weiss proved Poisson systems over many amenable groups were isomorphic in their 1987 paper. We consider Poisson systems over nondiscrete, noncompact, locally compact Polish groups, and we prove by construction all Poisson systems over such a group are finitarily isomorphic, producing examples of isomorphisms for nonamenable group actions. As a corollary, we prove Poisson systems and products of Poisson systems are finitarily isomorphic.

For a Poisson system over a group belonging to a slightly more restrictive class than above, we further prove it splits into two Poisson systems whose intensities sum to the intensity of the original, generalizing the same result for Poisson systems over Euclidean space proved by Holroyd, Lyons and Soo.

We establish a precise three-term asymptotic expansion, with an optimal estimate of the error term, for the rightmost eigenvalue of an $\mathit{n}\times \mathit{n}$ random matrix with independent identically distributed complex entries as n tends to infinity. All terms in the expansion are universal.

We prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope, which is uniform with respect to the choice of the graph. Moreover, we derive strict monotonicity properties for the probability of a wide class of “increasing” events, extending previous results of (Invent. Math. 188 (2012) 127–150). Our proof method is of independent interest and can be viewed as a reformulation of the ‘essential enhancements’ technique, which was introduced for percolation, in the framework of abelian networks.

The critical $2\mathit{d}$ stochastic heat flow (SHF) is a stochastic process of random measures on ${\mathbb{R}}^2$, recently constructed in (Invent. Math. 233 (2023) 325–460). We show that this process falls outside the class of Gaussian multiplicative chaos (GMC), in the sense that it cannot be realised as the exponential of a (generalised) Gaussian field. We achieve this by deriving strict lower bounds on the moments of the SHF that are of independent interest.

Place an A-particle at each site of a graph independently with probability p, and otherwise place a B-particle. A- and B-particles perform independent continuous time random walks at rates ${\mathit{\lambda }}_{\mathit{A}}$ and ${\mathit{\lambda }}_{\mathit{B}}$, respectively, and annihilate upon colliding with a particle of opposite type. Bramson and Lebowitz studied the setting ${\mathit{\lambda }}_{\mathit{A}}={\mathit{\lambda }}_{\mathit{B}}$ in the early 1990s. Despite recent progress, many basic questions remain unanswered when ${\mathit{\lambda }}_{\mathit{A}}\ne {\mathit{\lambda }}_{\mathit{B}}$. For the critical case $\mathit{p}=1/2$ on low-dimensional integer lattices, we give a lower bound on the expected number of particles at the origin that matches physicists’ predictions. For the process with ${\mathit{\lambda }}_{\mathit{B}}=0$ on the integers and on the bidirected regular tree, we give sharp upper and lower bounds for the expected total occupation time of the root at and approaching criticality.

We consider the N-particle Fleming–Viot process associated to a normally reflected diffusion with soft catalyst killing. The Fleming–Viot multicolor process is obtained by attaching genetic information to the particles in the Fleming–Viot process. We establish that, after rescaling time by $\mathit{t}\mapsto \mathit{N}\mathit{t}$, this genetic information converges to the (very different) Fleming–Viot process from population genetics, as $\mathit{N}\to \infty$. An extension is provided to dynamics given by Brownian motion with hard catalyst killing at the boundary of its domain.