We give an example of a long range Bernoulli percolation process on a group non-quasi-isometric with ℤ, in which clusters are almost surely finite for all values of the parameter. This random graph admits diverse equivalent definitions, and we study their ramifications. We also study its expected size and point out certain phase transitions.

Two main methods are used to solve continuous-time quasi birth-and-death processes: matrix geometric (MG) and probability generating functions (PGFs). MG requires a numerical solution (via successive substitutions) of a matrix quadratic equation A0 + RA1 + R2A2 = 0. PGFs involve a row vector $\vec{G}(z)$ of unknown generating functions satisfying $H(z)\vec{G}{(z)^\textrm{T}} = \vec{b}{(z)^\textrm{T}},$ where the row vector $\vec{b}(z)$ contains unknown “boundary” probabilities calculated as functions of roots of the matrix H(z). We show that: (a) H(z) and $\vec{b}(z)$ can be explicitly expressed in terms of the triple A0, A1, and A2; (b) when each matrix of the triple is lower (or upper) triangular, then (i) R can be explicitly expressed in terms of roots of $\det [H(z)]$; and (ii) the stability condition is readily extracted.

We introduce an idealised model for overland flow generated by rain falling on a hillslope. Our prime motivation is to show how the coalescence of runoff streams promotes the total generation of runoff. We show that, for our model, as the rate of rainfall increases in relation to the soil infiltration rate there is a distinct phase change. For low rainfall (the subcritical case) only the bottom of the hillslope contributes to the total overland runoff, while for high rainfall (the supercritical case) the whole slope contributes and the total runoff increases dramatically. We identify the critical point at which the phase change occurs, and show how it depends on the degree of coalescence. When there is no stream coalescence the critical point occurs when the rainfall rate equals the average infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the average infiltration rate, and increasing the amount of coalescence increases the total expected runoff.

We propose a wide generalization of known results related to the telegraph process. Functionals of the simple telegraph process on a straight line and their generalizations on an arbitrary state space are studied.

Using a result of Blanchet and Wallwater (2015) for exactly simulating the maximum of a negative drift random walk queue endowed with independent and identically distributed (i.i.d.) increments, we extend it to a multi-dimensional setting and then we give a new algorithm for simulating exactly the stationary distribution of a first-in–first-out (FIFO) multi-server queue in which the arrival process is a general renewal process and the service times are i.i.d.: the FIFO GI/GI/c queue with $ 2 \leq c \lt \infty$ . Our method utilizes dominated coupling from the past (DCFP) as well as the random assignment (RA) discipline, and complements the earlier work in which Poisson arrivals were assumed, such as the recent work of Connor and Kendall (2015). We also consider the models in continuous time, and show that with mild further assumptions, the exact simulation of those stationary distributions can also be achieved. We also give, using our FIFO algorithm, a new exact simulation algorithm for the stationary distribution of the infinite server case, the GI/GI/$\infty$ model. Finally, we even show how to handle fork–join queues, in which each arriving customer brings c jobs, one for each server.

We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernández and Procacci (2007), the proof is based on the Kirkwood–Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández, and Procacci (2010). In addition, we provide formulas for cumulants of double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.

In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit.

The frog model is a branching random walk on a graph in which particles branch only at unvisited sites. Consider an initial particle density of $\unicode[STIX]{x1D707}$ on the full $d$-ary tree of height $n$. If $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}(d^{2})$, all of the vertices are visited in time $\unicode[STIX]{x1D6E9}(n\log n)$ with high probability. Conversely, if $\unicode[STIX]{x1D707}=O(d)$ the cover time is $\exp (\unicode[STIX]{x1D6E9}(\sqrt{n}))$ with high probability.

Employing bijectivization of summation identities, we introduce local stochastic moves based on the Yang–Baxter equation for $U_{q}(\widehat{\mathfrak{sl}_{2}})$. Combining these moves leads to a new object which we call the spin Hall–Littlewood Yang–Baxter field—a probability distribution on two-dimensional arrays of particle configurations on the discrete line. We identify joint distributions along down-right paths in the Yang–Baxter field with spin Hall–Littlewood processes, a generalization of Schur processes. We consider various degenerations of the Yang–Baxter field leading to new dynamic versions of the stochastic six-vertex model and of the Asymmetric Simple Exclusion Process.

In this paper, we consider exponentiated location-scale model and obtain several ordering results between extreme order statistics in various senses. Under majorization type partial order-based conditions, the comparisons are established according to the usual stochastic order, hazard rate order and reversed hazard rate order. Multiple-outlier models are considered. When the number of components are equal, the results are obtained based on the ageing faster order in terms of the hazard rate and likelihood ratio orders. For unequal number of components, we develop comparisons according to the usual stochastic order, hazard rate order, and likelihood ratio order. Numerical examples are considered to illustrate the results.

We consider the nearest-neighbour simple exclusion process on the one-dimensional discrete torus $\mathbb{T}_N=\mathbb{Z}/N\mathbb{Z}$, with random rates $c_N=\{c_{x,N}\colon x \in \mathbb{T}_N\}$ defined in terms of a homogeneous Poisson process on $\mathbb{R}$ with intensity $\lambda$. Given a realization of the Poisson process, the jump rate along the edge $\{x,x+1\}$ is 1 if there is not any Poisson mark in $ (x,x+1) $ ; otherwise, it is $\lambda/N,\, \lambda \in( 0,1]$. The density profile of this process with initial measure associated to an initial profile $\rho_0\colon \mathbb{R}\rightarrow [0,1]$, evolves as the solution of a bounded diffusion random equation. This result follows from an appropriate quenched hydrodynamic limit. If $\lambda=1$ then $\rho$ is discontinuous at each Poisson mark with passage through the slow bonds, otherwise the conductance at the slow bonds decreases meaning no passage through the slow bonds in the continuum. The main results are concerned with upper and lower quenched and annealed bounds of $T_j$, where $T_j$ is the first displacement time of a tagged particle in a stochastic cluster of size j (the cluster is defined via specific macroscopic density profiles). It is possible to observe that when time t grows, then $\mathbb{P}\{T_j \geq t\}$ decays quadratically in both the upper and lower bounds, and falls as slow as the presence of more Poisson marks neighbouring the tagged particle, as expected.

The no restart random walk (NRRW) is a random network growth model driven by a random walk that builds the graph while moving on it, adding and connecting a new leaf node to the current position of the walker every s steps. We show a fundamental dichotomy in NRRW with respect to the parity of s: for ${s}=1$ we prove that the random walk is transient and non-leaf nodes have degrees bounded above by an exponential distribution; for s even we prove that the random walk is recurrent and non-leaf nodes have degrees bounded below by a power law distribution. These theoretical findings highlight and confirm the diverse and rich behaviour of NRRW observed empirically.

We study the impact of a random environment on lifetimes of coherent systems with dependent components. There are two combined sources of this dependence. One results from the dependence of the components of the coherent system operating in a deterministic environment and the other is due to dependence of components of the system sharing the same random environment. We provide different sets of sufficient conditions for the corresponding stochastic comparisons and consider various scenarios, namely, (i) two different (as a specific case, identical) coherent systems operate in the same random environment; (ii) two coherent systems operate in two different random environments; (iii) one of the coherent systems operates in a random environment and the other in a deterministic environment. Some examples are given to illustrate the proposed reasoning.

For a multivariate random walk with independent and identically distributed jumps satisfying the Cramér moment condition and having mean vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by and extends results of Avram et al. (2008) on a two-dimensional risk process. Our approach combines the large deviation techniques from a series of papers by Borovkov and Mogulskii from around 2000 with new auxiliary constructions, enabling us to extend their results on hitting remote sets with smooth boundaries to the case of boundaries with a ‘corner’ at the ‘most probable hitting point’. We also discuss how our results can be extended to the case of more general target sets.

This paper is devoted to studying the long-term behaviour of a continuous-time Markov chain that can be interpreted as a pair of linear birth processes which evolve with a competitive interaction; as a special case, they include the famous Lotka–Volterra interaction. Another example of our process is related to urn models with ball removal. We show that, with probability one, the process eventually escapes to infinity by sticking to the boundary in a rather unusual way.

We obtain a central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\unicode[STIX]{x1D714},x)=(\unicode[STIX]{x1D703}\unicode[STIX]{x1D714},T_{\unicode[STIX]{x1D714}}x)$ together with a $T$-invariant measure whose base map $\unicode[STIX]{x1D703}$ satisfies certain topological and mixing conditions and the maps $T_{\unicode[STIX]{x1D714}}$ on the fibers are certain non-singular distance-expanding maps. Our results hold true when $\unicode[STIX]{x1D703}$ is either a sufficiently fast mixing Markov shift with positive transition densities or a (non-uniform) Young tower with at least one periodic point and polynomial tails. The proofs are based on the random complex Ruelle–Perron–Frobenius theorem from Hafouta and Kifer [Nonconventional Limit Theorems and Random Dynamics. World Scientific, Singapore, 2018] applied with appropriate random transfer operators generated by $T_{\unicode[STIX]{x1D714}}$, together with certain regularity assumptions (as functions of $\unicode[STIX]{x1D714}$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition are also obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in Aimino, Nicol and Vaienti [Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Related Fields162 (2015), 233–274] does not seem to be applicable, and the dual of the Koopman operator of $T$ (with respect to the invariant measure) does not seem to have a spectral gap.

The Hamming graph H(d, n) is the Cartesian product of d complete graphs on n vertices. Let ${m=d(n-1)}$ be the degree and $V = n^d$ be the number of vertices of H(d, n). Let $p_c^{(d)}$ be the critical point for bond percolation on H(d, n). We show that, for $d \in \mathbb{N}$ fixed and $n \to \infty$,

$$p_c^{(d)} = {1 \over m} + {{2{d^2} - 1} \over {2{{(d - 1)}^2}}}{1 \over {{m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1/3}}),$$

$O(m^{-1}V^{-1/3})$

$d=4,5,6$

$m^{-3} =O(m^{-1}V^{-1/3})$

$p_c^{(d)}$

$d=2,3,4$

Consider any fixed graph whose edges have been randomly and independently oriented, and write {S ⇝} to indicate that there is an oriented path going from a vertex s ∊ S to vertex i. Narayanan (2016) proved that for any set S and any two vertices i and j, {S ⇝ i} and {S ⇝ j} are positively correlated. His proof relies on the Ahlswede–Daykin inequality, a rather advanced tool of probabilistic combinatorics.

In this short note I give an elementary proof of the following, stronger result: writing V for the vertex set of the graph, for any source set S, the events {S ⇝ i}, i ∊ V, are positively associated, meaning that the expectation of the product of increasing functionals of the family {S ⇝ i} for i ∊ V is greater than the product of their expectations.

We give a simple set of geometric conditions on curves $\unicode[STIX]{x1D702}$, $\widetilde{\unicode[STIX]{x1D702}}$ in $\mathbf{H}$ from $0$ to $\infty$ so that if $\unicode[STIX]{x1D711}:\mathbf{H}\rightarrow \mathbf{H}$ is a homeomorphism which is conformal off $\unicode[STIX]{x1D702}$ with $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702})=\widetilde{\unicode[STIX]{x1D702}}$ then $\unicode[STIX]{x1D711}$ is a conformal automorphism of $\mathbf{H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm–Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $\unicode[STIX]{x1D702}$ is a non-space-filling $\text{SLE}_{\unicode[STIX]{x1D705}}$ curve in $\mathbf{H}$ from $0$ to $\infty$, and $\unicode[STIX]{x1D711}$ is a homeomorphism which is conformal on $\mathbf{H}\setminus \unicode[STIX]{x1D702}$, and $\unicode[STIX]{x1D711}(\unicode[STIX]{x1D702})$, $\unicode[STIX]{x1D702}$ are equal in distribution, then $\unicode[STIX]{x1D711}$ is a conformal automorphism of $\mathbf{H}$. Applying this result for $\unicode[STIX]{x1D705}=4$ establishes that the welding operation for critical ($\unicode[STIX]{x1D6FE}=2$) LQG is well defined. Applying it for $\unicode[STIX]{x1D705}\in (4,8)$ gives a new proof that the welding of two independent $\unicode[STIX]{x1D705}/4$-stable looptrees of quantum disks to produce an $\text{SLE}_{\unicode[STIX]{x1D705}}$ on top of an independent $4/\sqrt{\unicode[STIX]{x1D705}}$-LQG surface is well defined.

This paper studies the variability of both series and parallel systems comprised of heterogeneous (and dependent) components. Sufficient conditions are established for the star and dispersive orderings between the lifetimes of parallel [series] systems consisting of dependent components having multiple-outlier proportional hazard rates and Archimedean [Archimedean survival] copulas. We also prove that, without any restriction on the scale parameters, the lifetime of a parallel or series system with independent heterogeneous scaled components is larger than that with independent homogeneous scaled components in the sense of the convex transform order. These results generalize some corresponding ones in the literature to the case of dependent scenarios or general settings of components lifetime distributions.