The $n$-dimensional binary hypercube is the graph whose vertices are the binary $n$-tuples $\{0,1\}^{n}$ and where two vertices are connected by an edge if they differ at exactly one coordinate. We prove that if the edges are assigned independent mean 1 exponential costs, the minimum length $T_{n}$ of a path from $(0,0,\dots,0)$ to $(1,1,\dots,1)$ converges in probability to $\ln(1+\sqrt{2})\approx0.881$. It has previously been shown by Fill and Pemantle [Ann. Appl. Probab. 3 (1993) 593–629] that this so-called first-passage time asymptotically almost surely satisfies $\ln(1+\sqrt{2})-o(1)\leq T_{n}\leq1+o(1)$, and has been conjectured to converge in probability by Bollobás and Kohayakawa [In Combinatorics, Geometry and Probability (Cambridge, 1993) (1997) 129–137 Cambridge]. A key idea of our proof is to consider a lower bound on Richardson’s model, closely related to the branching process used in the article by Fill and Pemantle to obtain the bound $T_{n}\geq\ln(1+\sqrt{2})-o(1)$. We derive an explicit lower bound on the probability that a vertex is infected at a given time. This result is formulated for a general graph and may be applicable in a more general setting.

Recently, there has been significant interest in understanding the properties of Markov random fields (M.r.f.) defined on the independent sets of sparse graphs. When these M.r.f. are restricted to pairwise interactions (i.e., hardcore model), much progress has been made. However, considerably less is known in the presence of higher-order interactions, which arise, for example, in the analysis of independent sets with special properties and the study of resource-constrained communication networks. In this paper, we further our understanding of such models by analyzing M.r.f. with second-order interactions on the independent sets of the infinite Cayley tree. We prove that the associated Gibbsian specification satisfies the celebrated FKG inequality whenever the local potentials defining the Hamiltonian satisfy a log-convexity condition. Under this condition, we give necessary and sufficient conditions for the existence of a unique infinite-volume Gibbs measure in terms of an explicit system of equations, prove the existence of a phase transition and give explicit bounds on the associated critical activity, which we prove to exhibit a certain robustness. For potentials which are small perturbations of those coinciding to the hardcore model at the critical activity, we characterize whether the resulting specification has a unique infinite-volume Gibbs measure in terms of whether these perturbations satisfy an explicit linear inequality. Our analysis reveals an interesting nonmonotonicity with regards to biasing toward excluded nodes with no included neighbors.

Exact approximations of Markov chain Monte Carlo (MCMC) algorithms are a general emerging class of sampling algorithms. One of the main ideas behind exact approximations consists of replacing intractable quantities required to run standard MCMC algorithms, such as the target probability density in a Metropolis–Hastings algorithm, with estimators. Perhaps surprisingly, such approximations lead to powerful algorithms which are exact in the sense that they are guaranteed to have correct limiting distributions. In this paper, we discover a general framework which allows one to compare, or order, performance measures of two implementations of such algorithms. In particular, we establish an order with respect to the mean acceptance probability, the first autocorrelation coefficient, the asymptotic variance and the right spectral gap. The key notion to guarantee the ordering is that of the convex order between estimators used to implement the algorithms. We believe that our convex order condition is close to optimal, and this is supported by a counterexample which shows that a weaker variance order is not sufficient. The convex order plays a central role by allowing us to construct a martingale coupling which enables the comparison of performance measures of Markov chain with differing invariant distributions, contrary to existing results. We detail applications of our result by identifying extremal distributions within given classes of approximations, by showing that averaging replicas improves performance in a monotonic fashion and that stratification is guaranteed to improve performance for the standard implementation of the Approximate Bayesian Computation (ABC) MCMC method.

In this paper, we study properties of solutions to stochastic differential equations with Sobolev diffusion coefficients and singular drifts. The properties we study include stability with respect to the coefficients, weak differentiability with respect to starting points and the Malliavin differentiability with respect to sample paths. We also establish Bismut–Elworthy–Li’s formula for the solutions. As an application, we use the stochastic Lagrangian representation of incompressible Navier–Stokes equations given by Constantin–Iyer [Comm. Pure Appl. Math. 61 (2008) 330–345] to prove the local well-posedness of NSEs in $\mathbb{R}^{d}$ with initial values in the first-order Sobolev space $\mathbb{W}^{1}_{p}(\mathbb{R}^{d};\mathbb{R}^{d})$ provided $p>d$.

We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2n\times2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that

\[\lim_{n\rightarrow\infty}\frac{1}{\sqrt{2n}}\log p_{2n,2k}=\lim_{n\rightarrow\infty}\frac{1}{\sqrt{2n}}\log p_{2n,0}=-\frac{1}{\sqrt{2\pi}}\zeta (\frac{3}{2}),\] where $\zeta$ is the Riemann zeta-function. Moreover, for any sequence of nonnegative integers $(k_{n})_{n\geq1}$,

\[\lim_{n\rightarrow\infty}\frac{1}{\sqrt{2n}}\log p_{2n,2k_{n}}=-\frac{1}{\sqrt{2\pi}}\zeta (\frac{3}{2}),\] provided $\lim_{n\rightarrow\infty}(n^{-1/2}\log(n))k_{n}=0$.

For a semi-martingale $X_{t}$, which forms a stochastic boundary, a rate-optimal estimator for its quadratic variation $\langle X,X\rangle_{t}$ is constructed based on observations in the vicinity of $X_{t}$. The problem is embedded in a Poisson point process framework, which reveals an interesting connection to the theory of Brownian excursion areas. We derive $n^{-1/3}$ as optimal convergence rate in a high-frequency framework with $n$ observations (in mean). We discuss a potential application for the estimation of the integrated squared volatility of an efficient price process $X_{t}$ from intra-day order book quotes.

Let $(M,g_{1})$ be a complete $d$-dimensional Riemannian manifold for $d>1$. Let $\mathcal{X}_{n}$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. We prove that the normalized length of the power-weighted shortest path between $x,y$ through $\mathcal{X}_{n}$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_{p}=f^{2(1-p)/d}g_{1}$, where $p>1$ is the power parameter.

We consider a stochastic model of infection spread incorporating monogamous partnership dynamics. In [Ann. Appl. Probab. 26 (2016) 1297–1328], a basic reproduction number $R_{0}$ is defined with the property that if $R_{0}<1$ the infection dies out within $O(\log N)$ units of time, while if $R_{0}>1$ the infection survives for at least $e^{\gamma N}$ units of time, for some $\gamma>0$. Here, we consider the critical case $R_{0}=1$ and show that the infection dies out within $O(\sqrt{N})$ units of time, and moreover that this estimate is sharp.

We investigate the asymptotic behavior as time goes to infinity of Hawkes processes whose regression kernel has $L^{1}$ norm close to one and power law tail of the form $x^{-(1+\alpha)}$, with $\alpha\in(0,1)$. We in particular prove that when $\alpha\in(1/2,1)$, after suitable rescaling, their law asymptotically behaves as a kind of integrated fractional Cox–Ingersoll–Ross process, with associated Hurst parameter $H=\alpha-1/2$. This result is in contrast to the case of a regression kernel with light tail, where a classical Brownian CIR process is obtained at the limit. Interestingly, it shows that persistence properties in the point process can lead to an irregular behavior of the limiting process. This theoretical result enables us to give an agent-based foundation to some recent findings about the rough nature of volatility in financial markets.

For $d\ge3$, we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $\mathbb{Z}^{d}$. This coupling has the advantage of working up to macroscopic subsets of the torus. As an application, we show a sharp phase transition for the diameter of the component of the vacant set on the torus containing a given point. The threshold where this phase transition takes place coincides with the critical value $u_{\star}(d)$ of random interlacements on $\mathbb{Z}^{d}$. Our main tool is a variant of the soft-local time coupling technique of Popov and Teixeira [J. Eur. Math. Soc. (JEMS) 17 (2015) 2545–2593].

We study networks of biochemical reactions modelled by continuous-time Markov processes. Such networks typically contain many molecular species and reactions and are hard to study analytically as well as by simulation. Particularly, we are interested in reaction networks with intermediate species such as the substrate-enzyme complex in the Michaelis–Menten mechanism. Such species are virtually in all real-world networks, they are typically short-lived, degraded at a fast rate and hard to observe experimentally.

We provide conditions under which the Markov process of a multiscale reaction network with intermediate species is approximated by the Markov process of a simpler reduced reaction network without intermediate species. We do so by embedding the Markov processes into a one-parameter family of processes, where reaction rates and species abundances are scaled in the parameter. Further, we show that there are close links between these stochastic models and deterministic ODE models of the same networks.

In this paper, a stochastic model of a large distributed system where users’ files are duplicated on unreliable data servers is investigated. Due to a server breakdown, a copy of a file can be lost, it can be retrieved if another copy of the same file is stored on other servers. In the case where no other copy of a given file is present in the network, it is definitively lost. In order to have multiple copies of a given file, it is assumed that each server can devote a fraction of its processing capacity to duplicate files on other servers to enhance the durability of the system.

A simplified stochastic model of this network is analyzed. It is assumed that a copy of a given file is lost at some fixed rate and that the initial state is optimal: each file has the maximum number $d$ of copies located on the servers of the network. The capacity of duplication policy is used by the files with the lowest number of copies. Due to random losses, the state of the network is transient and all files will be eventually lost. As a consequence, a transient $d$-dimensional Markov process $(X(t))$ with a unique absorbing state describes the evolution this network. By taking a scaling parameter $N$ related to the number of nodes of the network, a scaling analysis of this process is developed. The asymptotic behavior of $(X(t))$ is analyzed on time scales of the type $t\mapsto N^{p}t$ for $0\leq p\leq d-1$. The paper derives asymptotic results on the decay of the network: Under a stability assumption, the main results state that the critical time scale for the decay of the system is given by $t\mapsto N^{d-1}t$. In particular, the duration of time after which a fixed fraction of files are lost is of the order of $N^{d-1}$. When the stability condition is not satisfied, that is, when it is initially overloaded, it is shown that the state of the network converges to an interesting local equilibrium which is investigated. As a consequence, it sheds some light on the role of the key parameters $\lambda$, the duplication rate and $d$, the maximal number of copies, in the design of these systems. The techniques used involve careful stochastic calculus for Poisson processes, technical estimates and the proof of a stochastic averaging principle.

In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman–Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman–Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-square displacement of the underlying process which may serve as the theoretical foundation for the development of new scalable stochastic numerical homogenization schemes.

We obtain a perfect sampling characterization of weak ergodicity for backward products of finite stochastic matrices, and equivalently, simultaneous tail triviality of the corresponding nonhomogeneous Markov chains. Applying these ideas to hidden Markov models, we show how to sample exactly from the finite-dimensional conditional distributions of the signal process given infinitely many observations, using an algorithm which requires only an almost surely finite number of observations to actually be accessed. A notion of “successful” coupling is introduced and its occurrence is characterized in terms of conditional ergodicity properties of the hidden Markov model and related to the stability of nonlinear filters.

A random geometric irrigation graph $\Gamma_{n}(r_{n},\xi)$ has $n$ vertices identified by $n$ independent uniformly distributed points $X_{1},\ldots,X_{n}$ in the unit square $[0,1]^{2}$. Each point $X_{i}$ selects $\xi_{i}$ neighbors at random, without replacement, among those points $X_{j}$ ($j\neq i$) for which $\Vert X_{i}-X_{j}\Vert <r_{n}$, and the selected vertices are connected to $X_{i}$ by an edge. The number $\xi_{i}$ of the neighbors is an integer-valued random variable, chosen independently with identical distribution for each $X_{i}$ such that $\xi_{i}$ satisfies $\xi_{i}\ge1$. We prove that when $r_{n}=\gamma_{n}\sqrt{\log n/n}$ for $\gamma_{n}\to\infty$ with $\gamma_{n}=o(n^{1/6}/\log^{5/6}n)$, the random geometric irrigation graph experiences explosive percolation in the sense that if ${\mathbf{E} \xi_{i}=1}$, then the largest connected component has $o(n)$ vertices but if $\mathbf{E} \xi_{i}>1$, then the number of vertices of the largest connected component is, with high probability, $n-o(n)$. This offers a natural noncentralized sparsification of a random geometric graph that is mostly connected.

We consider Markovian multiclass multi-pool networks with heterogeneous server pools, each consisting of many statistically identical parallel servers, where the bipartite graph of customer classes and server pools forms a tree. Customers form their own queue and are served in the first-come first-served discipline, and can abandon while waiting in queue. Service rates are both class and pool dependent. The objective is to study the limiting diffusion control problems under the long run average (ergodic) cost criteria in the Halfin–Whitt regime. Two formulations of ergodic diffusion control problems are considered: (i) both queueing and idleness costs are minimized, and (ii) only the queueing cost is minimized while a constraint is imposed upon the idleness of all server pools. We develop a recursive leaf elimination algorithm that enables us to obtain an explicit representation of the drift for the controlled diffusions. Consequently, we show that for the limiting controlled diffusions, there always exists a stationary Markov control under which the diffusion process is geometrically ergodic. The framework developed in [Ann. Appl. Probab. 25 (2015) 3511–3570] is extended to address a broad class of ergodic diffusion control problems with constraints. We show that the unconstrained and constrained problems are well posed, and we characterize the optimal stationary Markov controls via HJB equations.

Given a survival distribution on the positive half-axis and a Brownian motion, a solution of the inverse first-passage problem consists of a boundary so that the first passage time over the boundary has the given distribution. We show that the solution of the inverse first-passage problem coincides with the solution of a related optimal stopping problem. Consequently, methods from optimal stopping theory may be applied in the study of the inverse first-passage problem. We illustrate this with a study of the associated integral equation for the boundary.

Markov chains can be used to generate samples whose distribution approximates a given target distribution. The quality of the samples of such Markov chains can be measured by the discrepancy between the empirical distribution of the samples and the target distribution. We prove upper bounds on this discrepancy under the assumption that the Markov chain is uniformly ergodic and the driver sequence is deterministic rather than independent $U(0,1)$ random variables. In particular, we show the existence of driver sequences for which the discrepancy of the Markov chain from the target distribution with respect to certain test sets converges with (almost) the usual Monte Carlo rate of $n^{-1/2}$.

In an Achlioptas process, starting with a graph that has $n$ vertices and no edge, in each round $d\ge1$ vertex pairs are chosen uniformly at random, and using some rule exactly one of them is selected and added to the evolving graph. We investigate the impact of the rule’s choice on one of the most basic properties of a graph: connectivity. In our main result we focus on the prominent class of bounded size rules, which select the edge to add according to the component sizes of its vertices, treating all sizes larger than some constant equally. For such rules we provide a fine analysis that exposes the limiting distribution and the expectation of the number of rounds until the graph gets connected, and we give a detailed picture of the dynamics of the formation of the single component from smaller components. Our results allow us to study the connectivity transition of all Achlioptas processes, in the sense that we identify a process that accelerates it as much as possible.