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We study the mixing time of the symmetric beta-binomial splitting process on finite weighted connected graphs $G=(V,E,\{r_e\}_{e\in E})$ with vertex set $V$, edge set $E$ and positive edge-weights $r_e>0$ for $e\in E$. This is an interacting particle system with a fixed number of particles that updates through vertex-pairwise interactions which redistribute particles. We show that the mixing time of this process can be upper-bounded in terms of the maximal expected meeting time of two independent random walks on $G$. Our techniques involve using a process similar to the chameleon process invented by Morris [2006] to bound the mixing time of the exclusion process.

Given a transition matrix $P$ indexed by a finite set $V$ of vertices, the voter model is a discrete-time Markov chain in $\{0,1\}^V$ where at each time-step a randomly chosen vertex $x$ imitates the opinion of vertex $y$ with probability $P(x,y)$. The noisy voter model is a variation of the voter model in which vertices may change their opinions by the action of an external noise. The strength of this noise is measured by an extra parameter $p \in [0,1]$. The noisy voter model is a Markov chain with state space $\{0,1\}^V$, and has a unique stationary distribution $\Gamma$ when $p > 0$. In this paper, we study $\Gamma$ when the number of vertices in $V$ tends to infinity. In particular we analyse the random variable $S = \sum_{x \in V}\pi(x)\xi(x)$, where $\pi$ is the stationary distribution of the transition matrix $P$, and $\xi \in \{0,1\}$ is sampled from $\Gamma$. Allowing $P$ and $p$ to be functions of the size $|V|$, we show that, under appropriate conditions, a normalised version of $S$ converges to a Gaussian random variable. We provide further analysis of the noisy voter model on a variety of graphs including the complete graph, cycle, torus and hypercube.

The Shapley value is one of the most widely used measures of feature importance partly as it measures a feature's average effect on a model's prediction. We introduce joint Shapley values, which directly extend Shapley's axioms and intuitions: joint Shapley values measure a set of features' average contribution to a model's prediction. We prove the uniqueness of joint Shapley values, for any order of explanation. Results for games show that joint Shapley values present different insights from existing interaction indices, which assess the effect of a feature within a set of features. The joint Shapley values provide intuitive results in ML attribution problems. With binary features, we present a presence-adjusted global value that is more consistent with local intuitions than the usual approach., Comment: Source code available at https://github.com/harris-chris/joint-shapley-values

We consider the interchange process with $k$ particles (${\rm IP}(k)$) on $n$-vertex hypergraphs in which each hyperedge $e$ rings at rate $r_e$. When $e$ rings, the particles occupying it are permuted according to a random permutation from some arbitrary law, where our only assumption is that ${\rm IP}(2)$ has uniform stationary distribution. We show that $t_{\rm mix}^{{\rm IP}(k)}(\epsilon)=O_{b}(t_{\rm mix}^{{\rm IP}(2)}(\epsilon/k))$, where $t_{\rm mix}^{{\rm IP}(i)}(\epsilon)$ is the $\epsilon$ total-variation mixing time of ${\rm IP}(i)$, provided that $kn^{-2}Rt_{\rm mix}^{{\rm IP}(2)}(\epsilon/k)=O((\epsilon/k)^b)$ for some $b>0$, where $R=\sum_e r_e|e|(|e|-1)$ is $n(n-1)$ times the particle-particle interaction rate at equilibrium. This has some consequences concerning the validity in this regime of conjectures of Oliveira about comparison of the $\epsilon$ mixing time of ${\rm IP}(k)$ to that of $k$ independent particles, each evolving according to ${\rm IP}(1)$, denoted ${\rm RW}(k)$, and of Caputo about comparison of the spectral-gap of ${\rm IP}(k)$ to that of a single particle ${\rm IP}(1)={\rm RW}(1)$. We also show that $t_{\rm mix}^{\mathrm{IP}(k)}(\epsilon) \asymp t_{\rm mix}^{{\rm RW}(1)}(\epsilon)\asymp t_{{\rm mix}}^{{\rm RW}(k)}(\epsilon k/4)$ for all $k\lesssim n^{1-\Omega(1)}$ and all $\epsilon\le\frac 1k\wedge\frac 14$ for vertex-transitive graphs of constant degree, as well as for general graphs satisfying a mild ("transience-like") heat-kernel condition. In the case where the particles occupying a hyperedge $e$ are permuted uniformly at random when $e$ rings we obtain results bounding the spectral gap of ${\rm IP}(k)$ in terms of that ${\rm RW}(1)$. The proof does not use Morris' chameleon process. It can be seen as a rigorous and direct way of arguing that when the number of particles is fairly small, the system behaves similarly to $k$ independent particles., Comment: 24 pages

Oliveira conjectured that the order of the mixing time of the exclusion process with $k$-particles on an arbitrary $n$-vertex graph is at most that of the mixing-time of $k$ independent particles. We verify this up to a constant factor for $d$-regular graphs when each edge rings at rate $1/d$ in various cases: (1) when $d = \Omega( \log_{n/k} n)$, (2) when $\mathrm{gap}:=$ the spectral-gap of a single walk is $ O ( 1/\log^4 n) $ and $k \ge n^{\Omega(1)}$, (3) when $k \asymp n^{a}$ for some constant $0

The Bouchaud-Anderson model (BAM) is a generalisation of the parabolic Anderson model (PAM) in which the driving simple random walk is replaced by a random walk in an inhomogeneous trapping landscape; the BAM reduces to the PAM in the case of constant traps. In this paper we study the BAM with double-exponential potential. We prove the complete localisation of the model whenever the distribution of the traps is unbounded. This may be contrasted with the case of constant traps (i.e. the PAM), for which it is known that complete localisation fails. This shows that the presence of an inhomogeneous trapping landscape may cause a system of branching particles to exhibit qualitatively distinct concentration behaviour., Comment: 42 pages

We investigate a variant of the parabolic Anderson model, introduced in previous work, in which an i.i.d.\! potential is partially duplicated in a symmetric way about the origin, with each potential value duplicated independently with a certain probability. In previous work we established a phase transition for this model on the integers in the case of Pareto distributed potential with parameter $\alpha > 1$ and fixed duplication probability $p \in (0, 1)$: if $\alpha \ge 2$ the model completely localises, whereas if $\alpha \in (1, 2)$ the model may localise on two sites. In this paper we prove a new phase transition in the case that $\alpha \ge 2$ is fixed but the duplication probability $p(n)$ varies with the distance from the origin. We identify a critical scale $p(n) \to 1$, depending on $\alpha$, below which the model completely localises and above which the model localises on exactly two sites. We further establish the behaviour of the model in the critical regime., Comment: 34 pages; published version

The parabolic Anderson model on $\mathbb{Z}^d$ with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small values of the exponent we show that the model may delocalise. More precisely, we show that there is an event of non-negligible probability on which the solution has non-negligible mass on two sites., Comment: Published version. 39 pages, 1 figure

We introduce a natural extension of the exclusion process to hypergraphs and prove an upper bound for its mixing time. In particular we show the existence of a constant $C$ such that for any connected, regular hypergraph $G$ within some natural class, the $\varepsilon$-mixing time of the exclusion process on $G$ with any feasible number of particles can be upper-bounded by $CT_{\text{EX}(2,G)}\log(|V|/\varepsilon)$, where $|V|$ is the number of vertices in $G$ and $T_{\text{EX}(2,G)}$ is the 1/4-mixing time of the corresponding exclusion process with just two particles. Moreover we show this is optimal in the sense that there exist hypergraphs in the same class for which $T_{\mathrm{EX}(2,G)}$ and the mixing time of just one particle are not comparable. The proofs involve an adaptation of the chameleon process, a technical tool invented by Morris and developed by Oliveira for studying the exclusion process on a graph.

Data obtained from ISSR amplification may readily be extracted but only allows us to know, for each gene, if a specific allele is present or not. From this partial information we provide a probabilistic method to reconstruct the pedigree corresponding to some families of diploid cultivars. This method consists in determining for each individual what is the most likely couple of parent pair amongst all older individuals, according to some probability measure. The construction of this measure bears on the fact that the probability to observe the specific alleles in the child, given the status of the parents does not depend on the generation and is the same for each gene. This assumption is then justified from a convergence result of gene frequencies which is proved here. Our reconstruction method is applied to a family of 85 living accessions representing the common broom {\it Cytisus scoparius}., Comment: 5 figures

It is well-known that both branching random walk models and trap models can exhibit intermittency and localisation phenomena; the prototypical examples being the parabolic Anderson and Bouchaud trap models respectively. Our aim is to investigate how these localisation phenomena interact. To do so, we study a hybrid model combining the dynamics of both the parabolic Anderson and the Bouchaud trap models; more precisely, we consider a variant of the parabolic Anderson model in which the underlying random walk driving the model is replaced with the Bouchaud trap model. In this initial study, we consider the model where the potential field distribution has Weibull tail decay and the trap distribution is bounded away from zero. In dimension one, we further assume that the trap distribution decays sufficiently fast. Under these conditions, we show that the localisation effects in the hybrid model are strongly dominated by the influence of the parabolic Anderson model; the Bouchaud trap model plays at most a minor role. Moreover, we distinguish regimes in which the Bouchaud trap model acts to strengthen or weaken the localisation effects due to the parabolic Anderson model, and also identify regimes in which the Bouchaud trap model has no influence at all on localisation in the hybrid model., Comment: Corrected typos, modified a coupling argument, updated references

Let $\sigma$ be a permutation of $\{0,\ldots,n\}$. We consider the Markov chain $X$ which jumps from $k\neq 0,n$ to $\sigma(k+1)$ or $\sigma(k-1)$, equally likely. When $X$ is at 0 it jumps to either $\sigma(0)$ or $\sigma(1)$ equally likely, and when $X$ is at $n$ it jumps to either $\sigma(n)$ or $\sigma(n-1)$, equally likely. We show that the identity permutation maximizes the expected hitting time of n, when the walk starts at 0. More generally, we prove that the hitting time of a random walk on a strongly connected $d$-directed graph is maximized when the graph is the line $[0,n]\cap\Z$ with $d-2$ self-loops at every vertex and $d-1$ self-loops at 0 and $n$.

We show that for any semi-random transposition shuffle on $n$ cards, the mixing time of any given $k$ cards is at most $n\log k$, provided $k=o((n/\log n)^{1/2})$. In the case of the top-to-random transposition shuffle we show that there is cutoff at this time with a window of size O(n), provided further that $k\to\infty$ as $n\to\infty$ (and no cutoff otherwise). For the random-to-random transposition shuffle we show cutoff at time $(1/2)n\log k$ for the same conditions on $k$. Finally, we analyse the cyclic-to-random transposition shuffle and show partial mixing occurs at time $\le\alpha n\log k$ for some $\alpha$ just larger than 1/2. We prove these results by relating the mixing time of $k$ cards to the mixing of one card. Our results rely heavily on coupling arguments to bound the total variation distance., Comment: 23 pages, 4 figures

We consider bond percolation on $n$ vertices on a circle where edges are permitted between vertices whose spacing is at most some number L=L(n). We show that the resulting random graph gets a giant component when $L\gg(\log n)^2$ (when the mean degree exceeds 1) but not when $L\ll\log n$. The proof uses comparisons to branching random walks. We also consider a related process of random transpositions of $n$ particles on a circle, where transpositions only occur again if the spacing is at most $L$. Then the process exhibits the mean-field behavior described by Berestycki and Durrett if and only if L(n) tends to infinity, no matter how slowly. Thus there are regimes where the random graph has no giant component but the random walk nevertheless has a phase transition. We discuss possible relevance of these results for a dataset coming from D. repleta and D. melanogaster and for the typical length of chromosomal inversions., Comment: Published in at http://dx.doi.org/10.1214/11-AAP793 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)