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Mark Holmes

Publications and source records attributed to Mark Holmes.

At least 19 recordsLinked to original sources
arXiv2025

Coalescence in Markov chains

A Markov chain $X^i$ on a finite state space $S$ has transition matrix $P$ and initial state $i$. We may run the chains $(X^i: i\in S)$ 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 $k(\mu)$ of coalescence classes of the process, and what is the set $K(P)$ of such numbers $k(\mu)$, as the coupling $\mu$ of the chains ranges over couplings that are consistent with $P$? We continue earlier work of the authors ('Non-coupling from the past', $\textit{In and Out of Equilibrium 3}$, Springer, 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.

math.PR↗
arXiv2025

Random skeletons in high-dimensional lattice trees

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 distributions 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.

math.PR↗
arXiv2024

All In: Give me your money!

We present a computer assisted proof for a result concerning a three player betting game, introduced by Angel and Holmes. The three players start with initial capital $x, y, z > 0$ respectively. At each step of this game two players are selected at random to bet on the outcome of a fair coin toss, with the size of the bet being the largest possible, namely the total capital held by the poorer of the two players at that time. The main quantity of interest is the probability of player 1 being eliminated (reaching 0 capital) first. Angel and Holmes have shown that this probability is not monotone decreasing as a function of the initial capital $x$ of player 1. They conjecture that if $x < y < z$ then player 1 would be better off (less likely to be eliminated first) by swapping their capital with another player. In this paper we present a computer-assisted proof of this conjecture. To achieve this, we introduce the theoretical framework MeshItUp, and then perform a two-stage reduction to make MeshItUp computationally feasible, through the use of mixed-integer programming.

math.PR↗
arXiv2024

Chemical distance for the half-orthant model

The half-orthant model is a partially oriented model of a random medium involving a parameter $p\in [0,1]$, 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 $\mathbb{Z}^d$. 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.

math.PR↗
arXiv2022

Percolation of terraces, and enhancements for the orthant model

We study a model of an i.i.d.~random environment in general dimensions $d\ge 2$, 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.

math.PR↗
arXiv2022

Cyclic products and optimal traps in cyclic birth and death chains

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.

math.PR↗
arXiv2022

Multi-purpose open-end monitoring procedures for multivariate observations based on the empirical distribution function

We propose nonparametric 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.

stat.ME↗
arXiv2021

Balanced Excited Random Walk in Two Dimensions

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

math.PR↗
arXiv2020

WARM percolation on a regular tree in the strong reinforcement regime

We consider a class of reinforcement processes, called WARMs, on tree graphs. These processes involve a parameter $\alpha$ which governs the strength of the reinforcement, and a collection of Poisson processes indexed by the vertices of the graph. It has recently been proved that for any fixed bounded degree graph with Poisson firing rates that are uniformly bounded above, in the very strong reinforcement regime ($\alpha\gg 1$ sufficiently large depending on the maximal degree), the set of edges that "survive" (i.e. that are reinforced infinitely often by the process) has only finite connected components. The present paper is devoted to the construction of an example in the opposite direction, that is, with the set of surviving edges having infinite connected components. Namely, we show that for each fixed $\alpha>1$ one can find a regular rooted tree and firing rates that are uniformly bounded from above, for which there are infinite components almost surely. Joining such examples, we find a graph (with unbounded degrees) on which for any $\alpha>1$ almost surely there are infinite connected components of surviving edges.

math.PR↗
arXiv2020

Open-end nonparametric sequential change-point detection based on the retrospective CUSUM statistic

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.

math.ST↗
arXiv2019

Phase transitions for degenerate random environments

We study a class of models of i.i.d.~random environments in general dimensions $d\ge 2$, where each site is equipped randomly with an environment, and a parameter $p$ governs the frequency of certain environments that can act as a barrier. We show that many of these models (including some which are non-monotone in $p$) exhibit a sharp phase transition for the geometry of connected clusters as $p$ varies.

math.PR↗
arXiv2019

A shape theorem for the orthant model

We study a particular model of a random medium, called the orthant model, in general dimensions $d\ge 2$. Each site $x\in \Z^d$ independently has arrows pointing to its positive neighbours $x+e_i$, $i=1,\dots, d$ with probability $p$ and otherwise to its negative neighbours $x-e_i$, $i=1,\dots, d$ (with probability $1-p$). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when $p$ is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary, which is a key requirement of the subadditive ergodic theorem.

math.PR↗
arXiv2019

Alignment percolation

The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d \ge 2$ 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 $p\in (0,1]$. 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.

math.PR↗
arXiv2019

Non-coupling from the past

The method of 'coupling from the past' permits exact sampling from the invariant distribution of a Markov chain on a finite state space. The coupling is successful whenever the stochastic dynamics are such that there is coalescence of all trajectories. The issue of the coalescence or non-coalescence of trajectories of a finite state space Markov chain is investigated in this note. The notion of the 'coalescence number' $k(\mu)$ of a Markovian coupling $\mu$ is introduced, and results are presented concerning the set $K(P)$ of coalescence numbers of couplings corresponding to a given transition matrix $P$. Note: This is a revision of the original published version, in which part of Theorem 6 has been removed. A correction may be found in Thm 5.3 of arXiv:2510.13572.

math.PR↗
arXiv2019

The continuous-time lace expansion

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 $g|\varphi|^4$ model on $\mathbb{Z}^{d}$ when $n=1,2$, and prove that the critical Green's function $G_{\nu_{c}}(x)$ is asymptotically a multiple of $|x|^{2-d}$ when $d\geq 5$ at weak coupling. As another application of our method we establish the analogous result for the lattice Edwards model at weak coupling.

math.PR↗
arXiv2019

Biased random walk on the trace of biased random walk on the trace of...

We study the behaviour of a sequence of biased random walks X(i), i>=0 on a sequence of random graphs, where the initial graph is Zd and otherwise the graph for the i-th walk is the trace of the (i - 1)-st walk. The sequence of bias vectors is chosen so that each walk is transient. We prove the aforementioned transience and a law of large numbers, and provide criteria for ballisticity and sub-ballisticity. We give examples of sequences of biases for which each X(i), i>=1 is (transient but) not ballistic, and the limiting graph is an infinite simple (self-avoiding) path. We also give examples for which each X(i), i>=1 is ballistic, but the limiting graph is not a simple path.

math.PR↗
arXiv2018

On the range of lattice models in high dimensions - extended version

In this paper we investigate the scaling limit of the range (the set of visited vertices) for a class of critical lattice models, starting from a single initial particle at the origin. We give conditions on the random sets and an associated "ancestral relation" under which, conditional on longterm survival, the rescaled ranges converge weakly to the range of super-Brownian motion as random sets. These hypotheses also give precise asymptotics for the limiting behaviour of exiting a large ball, that is for the extrinsic one-arm probabililty. We show that these conditions are satisfied by the voter model in dimensions $d \ge 2$ and critical sufficiently spread out lattice trees in dimensions $d > 8$. The latter result also has important consequences for the behaviour of random walks on lattice trees in high dimensions. We conjecture that our conditions are also satisfied by other models (at criticality above the critical dimension) such as sufficiently spread out oriented percolation and contact processes in dimensions d > 4. This version of the paper contains details not present in the submitted version.

math.PR↗