We consider a switched (queuing) network in which there are constraints on which queues may be served simultaneously; such networks have been used to effectively model input-queued switches and wireless networks. The scheduling policy for such a network specifies which queues to serve at any point in time, based on the current state or past history of the system. In the main result of this paper, we provide a new class of online scheduling policies that achieve optimal queue-size scaling for a class of switched networks including input-queued switches. In particular, it establishes the validity of a conjecture (documented in Shah, Tsitsiklis and Zhong [Queueing Syst. 68 (2011) 375–384]) about optimal queue-size scaling for input-queued switches.

This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a function of its summands as their number tends to infinity. In the large deviation range of the conditioning event it extends the Gibbs conditional principle in the sense that it provides a description of the distribution of the random walk on long subsequences. An approximation of the density of the runs is also obtained when the conditioning event states that the end value of the random walk belongs to a thin or a thick set with a nonempty interior. The approximations hold either in probability under the conditional distribution of the random walk, or in total variation norm between measures. An application of the approximation scheme to the evaluation of rare event probabilities through importance sampling is provided. When the conditioning event is in the range of the central limit theorem, it provides a tool for statistical inference in the sense that it produces an effective way to implement the Rao–Blackwell theorem for the improvement of estimators; it also leads to conditional inference procedures in models with nuisance parameters. An algorithm for the simulation of such long runs is presented, together with an algorithm determining the maximal length for which the approximation is valid up to a prescribed accuracy.

We provide simplified proofs for the asymptotic distribution of the number of cuts required to cut down a Galton–Watson tree with critical, finite-variance offspring distribution, conditioned to have total progeny $n$. Our proof is based on a coupling which yields a precise, nonasymptotic distributional result for the case of uniformly random rooted labeled trees (or, equivalently, Poisson Galton–Watson trees conditioned on their size). Our approach also provides a new, random reversible transformation between Brownian excursion and Brownian bridge.

Let $Z=(Z_{t})_{t\ge0}$ be a regular diffusion process started at $0$, let $\ell$ be an independent random variable with a strictly increasing and continuous distribution function $F$, and let $\tau_{\ell}=\inf\{t\ge0\vert Z_{t}=\ell\}$ be the first entry time of $Z$ at the level $\ell$. We show that the quickest detection problem

\[\inf_{\tau}[\mathsf{P}(\tau<\tau_{\ell})+c\mathsf{E}(\tau -\tau_{\ell})^{+}]\]

is equivalent to the (three-dimensional) optimal stopping problem

\[\sup_{\tau}\mathsf{E}[R_{\tau}-\int_{0}^{\tau}c(R_{t})\,dt],\]

where $R=S-I$ is the range process of $X=2F(Z)-1$ (i.e., the difference between the running maximum and the running minimum of $X$ ) and $c(r)=cr$ with $c>0$. Solving the latter problem we find that the following stopping time is optimal:

\[\tau_{*}=\inf \{t\ge0\vert f_{*}(I_{t},S_{t})\le X_{t}\le g_{*}(I_{t},S_{t})\},\]

where the surfaces $f_{*}$ and $g_{*}$ can be characterised as extremal solutions to a couple of first-order nonlinear PDEs expressed in terms of the infinitesimal characteristics of $X$ and $c$. This is done by extending the arguments associated with the maximality principle [Ann. Probab. 26 (1998) 1614–1640] to the three-dimensional setting of the present problem and disclosing the general structure of the solution that is valid in all particular cases. The key arguments developed in the proof should be applicable in similar multi-dimensional settings.

We study a physical system of $N$ interacting particles in $\mathbb{R}^{d}$, $d\geq1$, subject to pair repulsion and confined by an external field. We establish a large deviations principle for their empirical distribution as $N$ tends to infinity. In the case of Riesz interaction, including Coulomb interaction in arbitrary dimension $d>2$, the rate function is strictly convex and admits a unique minimum, the equilibrium measure, characterized via its potential. It follows that almost surely, the empirical distribution of the particles tends to this equilibrium measure as $N$ tends to infinity. In the more specific case of Coulomb interaction in dimension $d>2$, and when the external field is a convex or increasing function of the radius, then the equilibrium measure is supported in a ring. With a quadratic external field, the equilibrium measure is uniform on a ball.

We apply the objective method of Aldous to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and Wästlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the $\zeta(2)$ limit of the random assignment problem. A proof via the objective method is useful because it provides us with more information on the nature of the edge’s incident on a typical root in the minimum-cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This can be applied in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.

We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis–Hastings algorithm that adds an accept–reject mechanism to a Markov chain proposal in order to make the chain reversible with respect to the target measure. We focus on cases where the proposal is either a Gaussian random walk (RWM) with covariance equal to that of the reference measure or an Ornstein–Uhlenbeck proposal (pCN) for which the reference measure is invariant.

Previous results in terms of scaling and diffusion limits suggested that the pCN has a convergence rate that is independent of the dimension while the RWM method has undesirable dimension-dependent behaviour. We confirm this claim by exhibiting a dimension-independent Wasserstein spectral gap for pCN algorithm for a large class of target measures. In our setting this Wasserstein spectral gap implies an $L^{2}$-spectral gap. We use both spectral gaps to show that the ergodic average satisfies a strong law of large numbers, the central limit theorem and nonasymptotic bounds on the mean square error, all dimension independent. In contrast we show that the spectral gap of the RWM algorithm applied to the reference measures degenerates as the dimension tends to infinity.

This paper presents the asymptotic theory for nondegenerate $U$-statistics of high frequency observations of continuous Itô semimartingales. We prove uniform convergence in probability and show a functional stable central limit theorem for the standardized version of the $U$-statistic. The limiting process in the central limit theorem turns out to be conditionally Gaussian with mean zero. Finally, we indicate potential statistical applications of our probabilistic results.

Motivated by queues with many servers, we study Brownian steady-state approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steady-state, we prove, approximates that of the Markov chain with notable precision. Strong approximations provide such “limitless” approximations for process dynamics. Our focus here is on steady-state distributions, and the diffusion model that we propose is tractable relative to strong approximations.

Within an asymptotic framework, in which a scale parameter $n$ is taken large, a uniform (in the scale parameter) Lyapunov condition imposed on the sequence of diffusion models guarantees that the gap between the steady-state moments of the diffusion and those of the properly centered and scaled CTMCs shrinks at a rate of $\sqrt{n}$.

Our proofs build on gradient estimates for solutions of the Poisson equations associated with the (sequence of) diffusion models and on elementary martingale arguments. As a by-product of our analysis, we explore connections between Lyapunov functions for the fluid model, the diffusion model and the CTMC.

Consider a critical random multigraph $\mathcal{G}_{n}$ with $n$ vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second moment of $\nu$ is finite and equals twice its first moment). We specify the scaling limits of the ordered sequence of component sizes of $\mathcal{G}_{n}$ as $n$ tends to infinity in different cases. When $\nu$ has finite third moment, the components sizes rescaled by $n^{-2/3}$ converge to the excursion lengths of a Brownian motion with parabolic drift above past minima, whereas when $\nu$ is a power law distribution with exponent $\gamma\in(3,4)$, the components sizes rescaled by $n^{-(\gamma-2)/(\gamma-1)}$ converge to the excursion lengths of a certain nontrivial drifted process with independent increments above past minima. We deduce the asymptotic behavior of the component sizes of a critical random simple graph when $\nu$ has finite third moment.

We consider a class of discrete $q$-state spin models defined in terms of a translation-invariant quasilocal specification with discrete clock-rotation invariance which have extremal Gibbs measures $\mu'_{\varphi }$ labeled by the uncountably many values of $\varphi $ in the one-dimensional sphere (introduced by van Enter, Opoku, Külske [J. Phys. A 44 (2011) 475002, 11]). In the present paper we construct an associated Markov jump process with quasilocal rates whose semigroup $(S_{t})_{t\geq0}$ acts by a continuous rotation $S_{t}(\mu'_{\varphi })=\mu'_{\varphi +t}$.

As a consequence our construction provides examples of interacting particle systems with unique translation-invariant invariant measure, which is not long-time limit of all starting measures, answering an old question (compare Liggett [Interacting Particle Systems (1985) Springer], question four, Chapter one). The construction of this particle system is inspired by recent conjectures of Maes and Shlosman about the intermediate temperature regime of the nearest-neighbor clock model. We define our generator of the interacting particle system as a (noncommuting) sum of the rotation part and a Glauber part.

Technically the paper rests on the control of the spread of weak nonlocalities and relative entropy-methods, both in equilibrium and dynamically, based on Dobrushin-uniqueness bounds for conditional measures.