Search Results
A Gibbsian model for message routing in highly dense multi-hop networks
We investigate a probabilistic model for routing in relay-augmented multihop ad-hoc communication networks, where each user sends one message to the base station. Given the (random) user locations, we weigh the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured by how many pairs of hops use the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of a selfish optimization. We describe the joint routing strategy in terms of the empirical measure of all message trajectories. In the limit of high spatial density of users, we derive the limiting free energy and analyze the optimal strategy, given as the minimizer(s) of a characteristic variational formula. Interestingly, expressing the congestion term requires introducing an additional empirical measure.
On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion
Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions L that induce a flow, given by L(pt, pt) = 0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when L is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.
Large deviations for cluster size distributions in a continuous classical many-body system
An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.
Ideal mixture approximation of cluster size distributions at low density
We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture.
Hydrodynamic limit for the A + B → Ø model
We study a two-species interacting particle model on a subset of $Z$ with open boundaries. The two species are injected with time dependent rate on the left, resp. right boundary. Particles of different species annihilate when they try to occupy the same site. This model has been proposed as a simple model for the dynamics of an ``order book'' on a stock market. We consider the hydrodynamic scaling limit for the empirical process and prove a large deviation principle that implies convergence to the solution of a non-linear parabolic equation.
Brownian occupation measures, compactness and large deviations
In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=frac1tint_0^t1_A(W_s) d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $mathcal M_1(R^d)$ can be compactified by replacing the usual topology of weak c onvergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^d$ by adding a point at $infty$ that results in the compactification of $mathcal M_1(R^d)$ by allowing some mass to escape to the point at $infty$. If one were to use only test functions that are continuous and vanish at $infty$ then the compactification results in the space of sub-probability distributions $mathcal M_le 1(R^d)$ by ignoring the mass at $infty$. The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $widetildemathcal M_1=widetildemathcal M_1(R^d)$ under the action of the translation group $R^d$ on $mathcal M_1(R^d)$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.
Orthogonality of fluxes in general nonlinear reaction networks
We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.
Lower large deviations for geometric functionals
This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.
Large deviations for the local times of a random walk among random conductances
We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $Z^d$ in the spirit of Donsker-Varadhan citeDV75. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.
A large-deviations approach to gelation
A @large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t = 1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdos-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdos-Rényi graphs are connected.