Activated Random Walk is a particle system displaying Self-Organized Criticality, in that the dynamics spontaneously drive the system to a critical state. How universal is this critical state? We state many interlocking conjectures aimed at different aspects of this question: scaling limits, microscopic limits, temporal and spatial mixing, incompressibility, and hyperuniformity.

In this paper we study two notions of differentiability introduced by P. Cannarsa and G. Da Prato (see [28]) and L. Gross (see [56]) in both the framework of infinite dimensional analysis and the framework of Malliavin calculus.

We consider directed random graphs, the prototype of which being the Barak-Erdős graph $\overrightarrow{G}(\mathbb{Z},p)$, and study the way that long (or heavy, if weights are present) paths grow. This is done by relating the graphs to certain particle systems that we call Infinite Bin Models (IBM). A number of limit theorems are shown. The goal of this paper is to present results along with techniques that have been used in this area. In the case of $\overrightarrow{G}(\mathbb{Z},p)$ the last passage percolation constant $C(p)$ is studied in great detail. It is shown that $C(p)$ is analytic for $p>0$, has an interesting asymptotic expansion at $p=1$ and that $C(p)∕p$ converges to e like $1∕{(logp)}^2$ as $p\to 0$. The paper includes the study of IBMs as models on their own as well as their connections to stochastic models of branching processes in continuous or discrete time with selection. Several proofs herein are new or simplified versions of published ones. Regenerative techniques are used where possible, exhibiting random sets of vertices over which the graphs regenerate. When edges have random weights we show how the last passage percolation constants behave and when central limit theorems exist. When the underlying vertex set is partially ordered, new phenomena occur, e.g., there are relations with last passage Brownian percolation. We also look at weights that may possibly take negative values and study in detail some special cases that require combinatorial/graph theoretic techniques that exhibit some interesting non-differentiability properties of the last passage percolation constant. In addition, we explain how to approach the problem of estimation of last passage percolation constants by means of perfect simulation.

Partial Rejection Sampling is an algorithmic approach to obtaining a perfect sample from a specified distribution. The objects to be sampled are assumed to be represented by a number of random variables. In contrast to classical rejection sampling, in which all variables are resampled until a feasible solution is found, partial rejection sampling aims at greater efficiency by resampling only a subset of variables that ‘go wrong’. Partial rejection sampling is closely related to Moser and Tardos’ algorithmic version of the Lovász Local Lemma, but with the additional requirement that a specified output distribution should be met. This article provides a largely self-contained account of the basic form of the algorithm and its analysis. Working within a unified framework allows a clean expression of the running time, and clarifies the scope for nondeterminism in its implementation.

This introduction surveys a renormalisation group perspective on log-Sobolev inequalities and related properties of stochastic dynamics. We also explain the relationship of this approach to related recent and less recent developments such as Eldan’s stochastic localisation and the Föllmer process, the Boué–Dupuis variational formula and the Barashkov–Gubinelli approach, the transportation of measure perspective, and the classical analogues of these ideas for Hamilton–Jacobi equations which arise in mean-field limits.