We introduce a natural nonhierarchical version of Derrida’s generalized random energy model. We prove that, in the thermodynamical limit, the free energy is the same as that of a suitably constructed GREM.

We define the reflection of a random walk at a general barrier and derive, in case the increments are light tailed and have negative mean, a necessary and sufficient criterion for the global maximum of the reflected process to be finite a.s. If it is finite a.s., we show that the tail of the distribution of the global maximum decays exponentially fast and derive the precise rate of decay. Finally, we discuss an example from structural biology that motivated the interest in the reflection at a general barrier.

A deck of n cards is shuffled by repeatedly moving the top card to one of the bottom k_n positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as k_n ranges from a constant to n. We also consider a symmetric variant of this shuffle in which at each step either the top card is randomly inserted into the bottom k_n positions or a random card from the bottom k_n positions is moved to the top. For this reversible shuffle we derive bounds on the L² mixing time. Finally, we transfer mixing time estimates for the above shuffles to the lazy top to bottom-k walks that move with probability 1/2 at each step.

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavy-traffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called “interchange-of-limits” for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.

We obtain a new fluctuation identity for a general Lévy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766–1801] concerning asymptotic overshoot distribution of a particular class of Lévy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Lévy process is spectrally one sided.

The aim of this article is to provide a scheme for simulating diffusion processes evolving in one-dimensional discontinuous media. This scheme does not rely on smoothing the coefficients that appear in the infinitesimal generator of the diffusion processes, but uses instead an exact description of the behavior of their trajectories when they reach the points of discontinuity. This description is supplied with the local comparison of the trajectories of the diffusion processes with those of a skew Brownian motion.

We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward–backward SDEs, which provides an efficient probabilistic representation of this type of equation. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE. In particular, our work provides an alternative to the method described in [Douglas, Ma and Protter (1996) Ann. Appl. Probab. 6 940–968] and weakens the regularity assumptions required in this reference.

We address the problem of sampling colorings of a graph G by Markov chain simulation. For most of the article we restrict attention to proper q-colorings of a path on n vertices (in statistical physics terms, the one-dimensional q-state Potts model at zero temperature), though in later sections we widen our scope to general “H-colorings” of arbitrary graphs G. Existing theoretical analyses of the mixing time of such simulations relate mainly to a dynamics in which a random vertex is selected for updating at each step. However, experimental work is often carried out using systematic strategies that cycle through coordinates in a deterministic manner, a dynamics sometimes known as systematic scan. The mixing time of systematic scan seems more difficult to analyze than that of random updates, and little is currently known. In this article we go some way toward correcting this imbalance. By adapting a variety of techniques, we derive upper and lower bounds (often tight) on the mixing time of systematic scan. An unusual feature of systematic scan as far as the analysis is concerned is that it fails to be time reversible.

The overhand shuffle is one of the “real” card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37–49] who showed that the mixing time with respect to variation distance is at least of order n² and at most of order n²logn. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274–325] to establish a lower bound of order n²logn, thereby showing that n²logn is indeed the correct order of the mixing time. It is our hope that the extension of Wilson’s lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n³logn) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77–85] for the Rudvalis shuffle.

RED (Random Early Detection) has been suggested when multiple TCP sessions are multiplexed through a bottleneck buffer. The idea is to detect congestion before the buffer overflows by dropping or marking packets with a probability that increases with the queue length. The objectives are reduced packet loss, higher throughput, reduced delay and reduced delay variation achieved through an equitable distribution of packet loss and reduced synchronization.

Baccelli, McDonald and Reynier [Performance Evaluation 11 (2002) 77–97] have proposed a fluid model for multiple TCP connections in the congestion avoidance regime multiplexed through a bottleneck buffer implementing RED. The window sizes of each TCP session evolve like independent dynamical systems coupled by the queue length at the buffer. The key idea in [Performance Evaluation 11 (2002) 77–97] is to consider the histogram of window sizes as a random measure coupled with the queue. Here we prove the conjecture made in [Performance Evaluation 11 (2002) 77–97] that, as the number of connections tends to infinity, this system converges to a deterministic mean-field limit comprising the window size density coupled with a deterministic queue.

We consider the discrete Laplace operator Δ^(N) on Erdős–Rényi random graphs with N vertices and edge probability p/N. We are interested in the limiting spectral properties of Δ^(N) as N→∞ in the subcritical regime 0<p<1 where no giant cluster emerges. We prove that in this limit the expectation value of the integrated density of states of Δ^(N) exhibits a Lifshitz-tail behavior at the lower spectral edge E=0.

Consider the normalized partial sums of a real-valued function F of a Markov chain, $$\phi_{n}:=n^{-1}\sum_{k=0}^{n-1}F(\Phi(k)),\qquad n\ge1.$$ The chain {Φ(k):k≥0} takes values in a general state space $\mathsf {X}$, with transition kernel P, and it is assumed that the Lyapunov drift condition holds: $PV\le V-W+b\mathbb{I}_{C}$ where $V: \mathsf {X}\to(0,\infty)$, $W: \mathsf {X}\to[1,\infty)$, the set C is small and W dominates F. Under these assumptions, the following conclusions are obtained:

1. It is known that this drift condition is equivalent to the existence of a unique invariant distribution π satisfying π(W)<∞, and the law of large numbers holds for any function F dominated by W: $$ϕ_n→ϕ:=π(F),\qquad\mathrm{a.s.}, n→∞.$$

2. The lower error probability defined by $\mathsf {P}\{\phi_{n}\le c\}$, for c<ϕ, n≥1, satisfies a large deviation limit theorem when the function F satisfies a monotonicity condition. Under additional minor conditions an exact large deviations expansion is obtained.

3. If W is near-monotone, then control-variates are constructed based on the Lyapunov function V, providing a pair of estimators that together satisfy nontrivial large asymptotics for the lower and upper error probabilities.

In an application to simulation of queues it is shown that exact large deviation asymptotics are possible even when the estimator does not satisfy a central limit theorem.

The Generalized Pólya Urn (GPU) is a popular urn model which is widely used in many disciplines. In particular, it is extensively used in treatment allocation schemes in clinical trials. In this paper, we propose a sequential estimation-adjusted urn model (a nonhomogeneous GPU) which has a wide spectrum of applications. Because the proposed urn model depends on sequential estimations of unknown parameters, the derivation of asymptotic properties is mathematically intricate and the corresponding results are unavailable in the literature. We overcome these hurdles and establish the strong consistency and asymptotic normality for both the patient allocation and the estimators of unknown parameters, under some widely satisfied conditions. These properties are important for statistical inferences and they are also useful for the understanding of the urn limiting process. A superior feature of our proposed model is its capability to yield limiting treatment proportions according to any desired allocation target. The applicability of our model is illustrated with a number of examples.

A two-site spatial coagulation model is considered. Particles of masses m and n at the same site form a new particle of mass m+n at rate mn. Independently, particles jump to the other site at a constant rate. The limit (for increasing particle numbers) of this model is expected to be nondeterministic after the gelation time, namely, one or two giant particles randomly jump between the two sites. Moreover, a new effect of induced gelation is observed—the gelation happening at the site with the larger initial number of monomers immediately induces gelation at the other site. Induced gelation is shown to be of logarithmic order. The limiting behavior of the model is derived rigorously up to the gelation time, while the expected post-gelation behavior is illustrated by a numerical simulation.

Stochastic modeling of disease dynamics has had a long tradition. Among the first epidemic models including a spatial structure in the form of local interactions is the contact process. In this article we investigate two extensions of the contact process describing the course of a single disease within a spatially structured human population distributed in social clusters. That is, each site of the d-dimensional integer lattice is occupied by a cluster of individuals; each individual can be healthy or infected. The evolution of the disease depends on three parameters, namely the outside infection rate which models the interactions between the clusters, the within infection rate which takes into account the repeated contacts between individuals in the same cluster, and the size of each social cluster. For the first model, we assume cluster recoveries, while individual recoveries are assumed for the second one. The aim is to investigate the existence of nontrivial stationary distributions for both processes depending on the value of each of the three parameters. Our results show that the probability of an epidemic strongly depends on the recovery mechanism.

Let $\{X_{k,i};i≥1,k≥1\}$ be an array of i.i.d. random variables and let $\{p_n;n≥1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from $0$ and $∞$. For $W_n=\max _{1≤i<j≤p_n}|∑_{k=1}^nX_{k,i}X_{k,j}|$ and $L_n=\max _{1≤i<j≤p_n}|\hat{ρ}^{(n)}_{i,j}|$ where $\hat{ρ}^{(n)}_{i,j}$ denotes the Pearson correlation coefficient between $(X_{1,i},…,X_{n,i})'$ and $(X_{1,j},…,X_{n,j})'$, the limit laws (i) $\lim_{n\rightarrow \infty}\frac{W_{n}}{n^{\alpha}}=0$ a.s. $(α>1/2)$, (ii) $\lim _{n→∞}n^{1−α}L_n=0$ a.s. $(1/2<α≤1)$, (iii) $\lim_{n\rightarrow \infty}\frac{W_{n}}{\sqrt{n\log n}}=2$ a.s. and (iv) $\lim_{n\rightarrow \infty}(\frac{n}{\log n})^{1/2}L_{n}=2$ a.s. are shown to hold under optimal sets of conditions. These results follow from some general theorems proved for arrays of i.i.d. two-dimensional random vectors. The converses of the limit laws (i) and (iii) are also established. The current work was inspired by Jiang’s study of the asymptotic behavior of the largest entries of sample correlation matrices.

Mutualists and pathogens, collectively called symbionts, are ubiquitous in plant communities. While some symbionts are highly host-specific, others associate with multiple hosts. The outcomes of multispecies host-symbiont interactions with different degrees of specificity are difficult to predict at this point due to a lack of a general conceptual framework. Complicating our predictive power is the fact that plant populations are spatially explicit, and we know from past research that explicit space can profoundly alter plant-plant interactions. We introduce a spatially explicit, stochastic model to investigate the role of explicit space and host-specificity in multispecies host-symbiont interactions. We find that in our model, pathogens can significantly alter the spatial structure of plant communities, promoting coexistence, whereas mutualists appear to have only a limited effect. Effects are more pronounced the more host-specific symbionts are.