Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this paper, we present a unified approach to constructing stationary measures for most of the known colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (multi-species ASEP, or mASEP); (2) the $q$-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the $q$-Boson particle system; (3) the $q$-deformed Pushing Totally Asymmetric Simple Exclusion Process ($q$-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new ‘queue vertex models’ on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. For the mASEP on the ring, a particular case of our vertex model is equivalent to the multiline queues of Martin (Stationary distributions of the multi-type ASEP, Electron. J. Probab. 25 (2020), 1–41). For the colored $q$-Boson process and the $q$-PushTASEP on the ring, we recover and generalize known stationary measures constructed using multiline queues or other methods by Ayyer, Mandelshtam and Martin (Modified Macdonald polynomials and the multispecies zero range process: II, Algebr. Comb. 6 (2022), 243–284; Modified Macdonald polynomials and the multispecies zero-range process: I, Algebr. Comb. 6 (2023), 243–284) and Bukh and Cox (Periodic words, common subsequences and frogs, Ann. Appl. Probab. 32 (2022), 1295–1332). Our proofs of stationarity use the Yang–Baxter equation and bypass the Matrix Product Ansatz (used for the mASEP by Prolhac, Evans and Mallick (The matrix product solution of the multispecies partially asymmetric exclusion process, J. Phys. A. 42 (2009), 165004)). On the line and in a quadrant, we use the Yang–Baxter equation to establish a general colored Burke’s theorem, which implies that suitable specializations of our queue vertex models produce stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity.

In this paper, we prove that the hitting probability of the Minkowski sum of fractal percolations can be characterised by capacity. Then we extend this result to Minkowski sums of general random sets in $\mathbb Z^d$, including ranges of random walks and critical branching random walks, whose hitting probabilities are described by Newtonian capacity individually.

We prove a Poisson process approximation result for stabilising functionals of a determinantal point process. Our results use concrete couplings of determinantal processes with different Palm measures and exploit their association properties. Second, we focus on the Ginibre process and show in the asymptotic scenario of an increasing observation window that the process of points with a large nearest neighbour distance converges after a suitable scaling to a Poisson point process. As a corollary, we obtain the scaling of the maximum nearest neighbour distance in the Ginibre process, which turns out to be different from its analogue for independent points.

We analyze generating functions for trees and for connected subgraphs on the complete graph, and identify a single scaling profile which applies for both generating functions in a critical window. Our motivation comes from the analysis of the finite-size scaling of lattice trees and lattice animals on a high-dimensional discrete torus, for which we conjecture that the identical profile applies in dimensions $d \ge 8$.

We define a random graph obtained by connecting each point of $\mathbb{Z}^d$ independently and uniformly to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed k-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional k-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed k-neighbor graph between the vertices in at least one, respectively precisely two, directions. For these graphs we study the question of percolation, i.e. the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for $k=1$ even the undirected k-neighbor graph never percolates, while the directed k-neighbor graph percolates whenever $k \geq d+1$, $k \geq 3$, and $d \geq 5$, or $k \geq 4$ and $d=4$. We also show that the undirected 2-neighbor graph percolates for $d=2$, the undirected 3-neighbor graph percolates for $d=3$, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the k-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.

In their celebrated paper [CLR10], Caputo, Liggett and Richthammer proved Aldous’ conjecture and showed that for an arbitrary finite graph, the spectral gap of the interchange process is equal to the spectral gap of the underlying random walk. A crucial ingredient in the proof was the Octopus Inequality — a certain inequality of operators in the group ring $\mathbb{R}\left[{\mathrm{Sym}}_{n}\right]$ of the symmetric group. Here we generalise the Octopus Inequality and apply it to generalising the Caputo–Liggett–Richthammer Theorem to certain hypergraphs, proving some cases of a conjecture of Caputo.

We consider the hard-core model on a finite square grid graph with stochastic Glauber dynamics parametrized by the inverse temperature $\beta$. We investigate how the transition between its two maximum-occupancy configurations takes place in the low-temperature regime $\beta \to \infty$ in the case of periodic boundary conditions. The hard-core constraints and the grid symmetry make the structure of the critical configurations for this transition, also known as essential saddles, very rich and complex. We provide a comprehensive geometrical characterization of these configurations that together constitute a bottleneck for the Glauber dynamics in the low-temperature limit. In particular, we develop a novel isoperimetric inequality for hard-core configurations with a fixed number of particles and show how the essential saddles are characterized not only by the number of particles but also their geometry.

We consider interacting urns on a finite directed network, where both sampling and reinforcement processes depend on the nodes of the network. This extends previous research by incorporating node-dependent sampling and reinforcement. We classify the sampling and reinforcement schemes, as well as the networks on which the proportion of balls of either colour in each urn converges almost surely to a deterministic limit. We also investigate conditions for achieving synchronisation of the colour proportions across the urns and analyse fluctuations under specific conditions on the reinforcement scheme and network structure.

We introduce and study a game-theoretic model to understand the spread of an epidemic in a homogeneous population. A discrete-time stochastic process is considered where, in each epoch, first, a randomly chosen agent updates their action trying to maximize a proposed utility function, and then agents who have viral exposures beyond their immunity get infected. Our main results discuss asymptotic limiting distributions of both the cardinality of the subset of infected agents and the action profile, considered under various values of two parameters (initial action and immunity profile). We also show that the theoretical distributions are almost always achieved in the first few epochs.

The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this paper, we consider a multicolor box-ball system with two types of random initial configurations and obtain sharp scaling limits of the soliton lengths as the system size tends to infinity. We obtain a sharp scaling limit of soliton lengths that turns out to be more delicate than that in the single color case established in [LLP20]. A large part of our analysis is devoted to studying the associated carrier process, which is a multidimensional Markov chain on the orthant, whose excursions and running maxima are closely related to soliton lengths. We establish the sharp scaling of its ruin probabilities, Skorokhod decomposition, strong law of large numbers and weak diffusive scaling limit to a semimartingale reflecting Brownian motion with explicit parameters. We also establish and utilize complementary descriptions of the soliton lengths and numbers in terms of modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes.

We consider the constrained-degree percolation model in a random environment (CDPRE) on the square lattice. In this model, each vertex v has an independent random constraint $\kappa_v$ which takes the value $j\in \{0,1,2,3\}$ with probability $\rho_j$. The dynamics is as follows: at time $t=0$ all edges are closed; each edge e attempts to open at a random time $U(e)\sim \mathrm{U}(0,1]$, independently of all the other edges. It succeeds if at time U(e) both its end vertices have degrees strictly smaller than their respective constraints. We obtain exponential decay of the radius of the open cluster of the origin at all times when its expected size is finite. Since CDPRE is dominated by Bernoulli percolation, this result is meaningful only if the supremum of all values of t for which the expected size of the open cluster of the origin is finite is larger than $\frac12$. We prove this last fact by showing a sharp phase transition for an intermediate model.

We answer the following question: if the occupied (or vacant) set of a planar Poisson Boolean percolation model contains a crossing of an $n\times n$ square, how wide is this crossing? The answer depends on whether we consider the critical, sub-, or super-critical regime, and is different for the occupied and vacant sets.

We introduce gradient flow aggregation, a random growth model. Given existing particles $\{x_1,\ldots,x_n\} \subset \mathbb{R}^2$, a new particle arrives from a random direction at $\infty$ and flows in direction of the vector field $\nabla E$ where $ E(x) = \sum_{i=1}^{n}{1}/{\|x-x_i\|^{\alpha}}$, $0 < \alpha < \infty$. The case $\alpha = 0$ refers to the logarithmic energy ${-}\sum\log\|x-x_i\|$. Particles stop once they are at distance 1 from one of the existing particles, at which point they are added to the set and remain fixed for all time. We prove, under a non-degeneracy assumption, a Beurling-type estimate which, via Kesten’s method, can be used to deduce sub-ballistic growth for $0 \leq \alpha < 1$, $\text{diam}(\{x_1,\ldots,x_n\}) \leq c_{\alpha} \cdot n^{({3 \alpha +1})/({2\alpha + 2})}$. This is optimal when $\alpha = 0$. The case $\alpha = 0$ leads to a ‘round’ full-dimensional tree. The larger the value of $\alpha$, the sparser the tree. Some instances of the higher-dimensional setting are also discussed.

We derive an asymptotic expansion for the critical percolation density of the random connection model as the dimension of the encapsulating space tends to infinity. We calculate rigorously the first expansion terms for the Gilbert disk model, the hyper-cubic model, the Gaussian connection kernel, and a coordinate-wise Cauchy kernel.

We study the low-temperature $(2+1)$D solid-on-solid model on with zero boundary conditions and nonnegative heights (a floor at height $0$). Caputo et al. (2016) established that this random surface typically admits either $\mathfrak h $ or $\mathfrak h+1$ many nested macroscopic level line loops $\{\mathcal L_i\}_{i\geq 0}$ for an explicit $\mathfrak h\asymp \log L$, and its top loop $\mathcal L_0$ has cube-root fluctuations: For example, if $\rho (x)$ is the vertical displacement of $\mathcal L_0$ from the bottom boundary point $(x,0)$, then $\max \rho (x) = L^{1/3+o(1)}$ over . It is believed that rescaling $\rho $ by $L^{1/3}$ and $I_0$ by $L^{2/3}$ would yield a limit law of a diffusion on $[-1,1]$. However, no nontrivial lower bound was known on $\rho (x)$ for a fixed $x\in I_0$ (e.g., $x=\frac L2$), let alone on $\min \rho (x)$ in $I_0$, to complement the bound on $\max \rho (x)$. Here, we show a lower bound of the predicted order $L^{1/3}$: For every $\epsilon>0$, there exists $\delta>0$ such that $\min _{x\in I_0} \rho (x) \geq \delta L^{1/3}$ with probability at least $1-\epsilon $. The proof relies on the Ornstein–Zernike machinery due to Campanino–Ioffe–Velenik and a result of Ioffe, Shlosman and Toninelli (2015) that rules out pinning in Ising polymers with modified interactions along the boundary. En route, we refine the latter result into a Brownian excursion limit law, which may be of independent interest. We further show that in a $ K L^{2/3}\times K L^{2/3}$ box with boundary conditions $\mathfrak h-1,\mathfrak h,\mathfrak h,\mathfrak h$ (i.e., $\mathfrak h-1$ on the bottom side and $\mathfrak h$ elsewhere), the limit of $\rho (x)$ as $K,L\to \infty $ is a Ferrari–Spohn diffusion.

The binary contact path process (BCPP) introduced in Griffeath (1983) describes the spread of an epidemic on a graph and is an auxiliary model in the study of improving upper bounds of the critical value of the contact process. In this paper, we are concerned with limit theorems of the occupation time of a normalized version of the BCPP (NBCPP) on a lattice. We first show that the law of large numbers of the occupation time process is driven by the identity function when the dimension of the lattice is at least 3 and the infection rate of the model is sufficiently large conditioned on the initial state of the NBCPP being distributed with a particular invariant distribution. Then we show that the centered occupation time process of the NBCPP converges in finite-dimensional distributions to a Brownian motion when the dimension of the lattice and the infection rate of the model are sufficiently large and the initial state of the NBCPP is distributed with the aforementioned invariant distribution.

We investigate the hyperuniformity of marked Gibbs point processes that have weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Various stability and range assumptions are imposed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models, including Gibbs point processes with a superstable, lower-regular, integrable pair potential, as well as the Widom–Rowlinson model with random radii and Gibbs point processes with interactions based on Voronoi tessellations and nearest-neighbour graphs.

Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on $\mathbb{R}^d$. A graph $\mathcal{G}_\mathcal{V}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points (v, u) with $v\in\mathcal{V}$ and $u\in\mathcal{U}$ independently with probability $g(v-u)$, where g is a non-increasing radial function, and then connecting two points $v_1,v_2\in\mathcal{V}$ if and only if they have a joint neighbor $u\in\mathcal{U}$. This gives rise to a random intersection graph on $\mathbb{R}^d$. Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function g. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether g has bounded or unbounded support.

We define the co-spectral radius of inclusions ${\mathcal S}\leq {\mathcal R}$ of discrete, probability- measure-preserving equivalence relations as the sampling exponent of a generating random walk on the ambient relation. The co-spectral radius is analogous to the spectral radius for random walks on $G/H$ for inclusion $H\leq G$ of groups. For the proof, we develop a more general version of the 2–3 method we used in another work on the growth of unimodular random rooted trees. We use this method to show that the walk growth exists for an arbitrary unimodular random rooted graph of bounded degree. We also investigate how the co-spectral radius behaves for hyperfinite relations, and discuss new critical exponents for percolation that can be defined using the co-spectral radius.

We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in $\{1,\dots,n\}$ occurs k times, where k may depend on n. This generalises the famous Ulam–Hammersley problem of the case $k=1$. The proof relies on poissonisation and on a careful non-asymptotic analysis of variants of the Hammersley–Aldous–Diaconis particle system.