A Markov chain X(i )on a finite state space S has transition matrix P and initial state i. We may run the chains (X-i : i is an element of S) in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are Strajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number k(& micro;) of coalescence classes of the process, and what is the set K (P) of such numbers k(& micro;), as the coupling & micro; of the chains ranges over couplings that are consistent with P? We continue earlier work of Grimmett and Holmes (In: In and out of equilibrium 3, Birkhauser/Springer, Cham, 2021) on these two fundamental questions, which have special importance for the "coupling from the past" algorithm. We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented and also of non-block measure with similar properties.

A Markov chain X^i on a finite state space S has transition matrix P and initial state i. We may run the chains (X^i: i∈ S) in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are |S| trajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number k(μ ) of coalescence classes of the process, and what is the set K(P) of such numbers k(μ ) , as the coupling μ of the chains ranges over couplings that are consistent with P? We continue earlier work of Grimmett and Holmes (In: In and out of equilibrium 3, Birkhäuser/Springer, Cham, 2021) on these two fundamental questions, which have special importance for the “coupling from the past” algorithm. We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented and also of non-block measure with similar properties.

We study the behaviour of the rescaled minimal subtree containing the origin and K random vertices selected from a random critical (sufficiently spread-out, and in dimensions d > 8) lattice tree conditioned to survive until time ns, in the limit as n goes to infinity. We prove joint weak convergence of various quantities associated with these subtrees under this sequence of conditional measures to their counterparts for historical Brownian motion. We also show that when K is sufficiently large the entire rescaled tree is close to this rescaled skeleton with high probability, uniformly in n. These two results are the key conditions used in [5] to prove that the simple random walk on sufficiently spread-out lattice trees (conditioned to survive for a long time) converges to Brownian motion on a super-Brownian motion (conditioned to survive). The main convergence result is established more generally for a sequence of historical processes converging to historical Brownian motion in the sense of finite dimensional distri- butions and satisfying a pair of technical conditions. The conditions are readily verified for the lattice trees mentioned above and also for critical branching random walk. We expect that it will also apply with suitable changes to other lattice models in sufficiently high dimensions such as oriented percolation and the voter model. In addition some forms of the second skeleton density result are already established in this generality.

We study a class of discrete geometric objects that we call terraces. These objects arise in, and are crucial for the study of phase transitions for a certain model of a random environment in general dimensions d ≥ 2 d\ge 2 . In this model, each lattice site is equipped with one of two local environments, with a parameter p p governing the frequency of the first local environment. For each dimension d d there is a critical parameter p c ( d ) p_c(d) at which a phase transition occurs for the connected cluster of sites that can be reached from the origin. We prove various results about local deformations of terraces, and subsequently apply the celebrated percolation theoretic methodology of enhancements in this novel setting to prove that p c ( d ) p_c(d) is strictly monotone in d d for this model.

The half-orthant model is a partially oriented model of a random medium involving a parameter p∈ [0,1], for which there is a critical value p_c(d) (depending on the dimension d) below which every point is reachable from the origin. We prove a limit theorem for the graph-distance (or "chemical distance") for this model when p<p_c(2), and also when 1-p is larger than the critical parameter for site percolation in ℤ^d. The proof involves an application of the subadditive ergodic theorem. Novel arguments herein include the method of proving that the expected number of steps to reach any given point is finite, as well as an argument that is used to show that the shape is "non-trivial" in certain directions.

We propose non-parametric open-end sequential testing procedures that can detect all types of changes in the contemporary distribution function of possibly multivariate observations. Their asymptotic properties are theoretically investigated under stationarity and under alternatives to stationarity. Monte Carlo experiments reveal their good finite-sample behavior in the case of continuous univariate, bivariate and trivariate observations. A short data example concludes the work.

Stochastic processes with reinforcement are the central theme of the present tandem workshop. We assembled a diverse group of international experts that worked on reinforcement dynamics from several different perspectives. We discussed progress and future strategies around a number of key open problems in the area of interacting urns with graph based interaction, preferential attachment, and reinforced random walks.

We study a reinforcement process on graphs G of bounded degree. The model involves a parameter alpha> 0 governing the strength of reinforcement, and Poisson clock rates lambda(v) at the vertices nu of the graph. When the Poisson clock at a vertex v rings, one of the edges incident to it is reinforced, with edge e being chosen with probability proportional to its current count (counts start from 1) raised to the power a. The main problem in such models is to describe the (random) subgraph E-infinity, consisting of edges that are reinforced infinitely often. In this paper, we focus on the finite connected components of E 8 in the strong reinforcement regime (alpha> 1) with clock rates that are uniformly bounded above. We show here that when a is sufficiently large, all connected components of E-infinity are trees. When the firing rates lambda(v) are constant, we show that all components are trees of diameter at most 3 when alpha is sufficiently large, and that there are infinitely many phase transitions as alpha down arrow 1. For example, on the triangular lattice, increasingly large (odd) cycles appear as alpha down arrow 1 (while on the square lattice no finite component of E-infinity contains a cycle for any alpha> 1). Increasingly long paths and other structures appear in both lattices when taking alpha down arrow 1. In the special case where G = Z and alpha> 1, all connected components of E-infinity are finite and we show that the possible cluster sizes are non-monotone in alpha. We also present several open problems.

We give an explicit formula for the mean square displacement of the random walk on the d-dimensional Manhattan lattice after n steps, for all n and all dimensions d≥2.

We give non-trivial upper and lower bounds on the range of the so-called Balanced Excited Random Walk in two dimensions, and verify a conjecture of Benjamini, Kozma and Schapira. To the best of our knowledge these are the first non-trivial results for this 2-dimensional model

We prove that the rescaled historical processes associated to critical spread-out lattice trees in dimensions d>8 converge to historical Brownian motion. This is a functional limit theorem for measure-valued processes that encodes the genealogical structure of the underlying random trees. Our results are applied elsewhere to prove that random walks on lattice trees, appropriately rescaled, converge to Brownian motion on super-Brownian motion.

A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.

We analyse a generalisation of the stochastic gene expression model studied recently in Fromion et al. (SIAM J Appl Math 73:195–211, 2013) and Robert (Probab Surv 16:277–332, 2019) that keeps track of the production of both mRNA and protein molecules, using techniques from the theory of point processes, as well as ideas from the theory of matrix-analytic methods. Here, both the activity of a gene and the creation of mRNA are modelled with an arbitrary Markovian Arrival Process governed by finitely many phases, and each mRNA molecule during its lifetime gives rise to protein molecules in accordance with a Poisson process. This modification is important, as Markovian Arrival Processes can be used to approximate many types of point processes on the nonnegative real line, meaning this framework allows us to further relax our assumptions on the overall process of transcription.

We study the so-called frog model on Z with two types of lazy frogs, with parameters p(1), p(2) is an element of (0, 1] respectively, and a finite expected number of dormant frogs per site. We show that for any such p(1) and p(2) there is positive probability that the two types coexist (i.e. that both types activate infinitely many frogs). This answers a question of Deijfen, Hirscher, and Lopes in dimension one.

We study a model of an i.i.d. random environment in general dimensions d≥ 2, where each site is equipped with one of two environments. The model comes with a parameter p which governs the frequency of the first environment, and for each dimension d there is a critical parameter p_c(d) at which there is a phase transition for the geometry of a particular connected cluster (the cluster is infinite for all p). We use the celebrated methodology of enhancements in this novel setting to prove that p_c(d) is strictly monotone in d for this model. To do so we study the discrete geometry and percolation theory of higher-dimensional structures called terraces.

Abstract We study the so-called frog model on ${\mathbb{Z}}$ with two types of lazy frogs, with parameters $p_1,p_2\in (0,1]$ respectively, and a finite expected number of dormant frogs per site. We show that for any such $p_1$ and $p_2$ there is positive probability that the two types coexist (i.e. that both types activate infinitely many frogs). This answers a question of Deijfen, Hirscher, and Lopes in dimension one.

We derive a continuous-time lace expansion for a broad class of self-interacting continuous-time random walks. Our expansion applies when the self-interaction is a sufficiently nice function of the local time of a continuous-time random walk. As a special case we obtain a continuous-time lace expansion for a class of spin systems that admit continuous-time random walk representations. We apply our lace expansion to the n-component g|φ|4 model on Zd when n = 1, 2, and prove that the critical Green’s function Gνc(x) is asymptotically a multiple of |x| when d ≥ 5 at weak coupling. As another application of our method we establish the analogous result for the lattice Edwards model at weak coupling.

We study a class of reinforcement models involving a Poisson process on the vertices of certain infinite graphs G. When a vertex fires, one of the edges incident to that vertex is selected. The edge selection is biased towards edges that have been selected many times previously, and a parameter a governs the strength of this bias. We show that for various graphs (including all graphs of bounded degree), if alpha >> 1 (the very strong reinforcement regime) then the random subgraph consisting of edges that are ever selected by this process does not percolate (all connected components are finite). Combined with results appearing in a companion paper, this proves that on these graphs, with alpha sufficiently large, all connected components are in fact trees. If the Poisson firing rates are constant over the vertices, then these trees are of diameter at most 3. The proof of nonpercolation relies on coupling with a percolation-type model that may be of interest in its own right.

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d \geqslant 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on $\mathbb {Z}^{d}$ with parameter p ∈ (0,1]. For each occupied site v, and for each of the 2d possible coordinate directions, declare the entire line segment from v to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the ‘one-choice model’, each occupied site declares one of its 2d incident segments to be blue. In the ‘independent model’, the states of different line segments are independent.

The aim of online monitoring is to issue an alarm as soon as there is significant evidence in the collected observations to suggest that the underlying data generating mechanism has changed. This work is concerned with open-end, nonparametric procedures that can be interpreted as statistical tests. The proposed monitoring schemes consist of computing the so-called retrospective CUSUM statistic (or minor variations thereof) after the arrival of each new observation. After proposing suitable threshold functions for the chosen detectors, the asymptotic validity of the procedures is investigated in the special case of monitoring for changes in the mean, both under the null hypothesis of stationarity and relevant alternatives. To carry out the sequential tests in practice, an approach based on an asymptotic regression model is used to estimate high quantiles of relevant limiting distributions. Monte Carlo experiments demonstrate the good finite-sample behavior of the proposed monitoring schemes and suggest that they are superior to existing competitors as long as changes do not occur at the very beginning of the monitoring. Extensions to statistics exhibiting an asymptotic mean-like behavior are briefly discussed. Finally, the application of the derived sequential change-point detection tests is succinctly illustrated on temperature anomaly data.