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- ItemA Gibbsian model for message routing in highly dense multi-hop networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) König, Wolfgang; Tóbiás, AndrásWe 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.
- ItemBrownian occupation measures, compactness and large deviations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mukherjee, Chiranjib; Varadhan, S.R. SrinivasaIn 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.
- ItemA large-deviations approach to gelation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Andreis, Luisa; König, Wolfgang; Patterson, RobertA @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.
- ItemBrownian occupation measures, compactness and large deviations: Pair interaction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Mukherjee, ChiranjibContinuing with the study of compactness and large deviations initiated in citeMV14, we turn to the analysis of Gibbs measures defined on two independent Brownian paths in $R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $intint_R^2d V(x-y) mu(d x)nu(d y)$ with two probability measures $mu$ and $nu$ representing the occupation measures of two independent Brownian motions. Due to the mixed product of two independent measures, the crucial shift-invariance requirement of citeMV14 is slightly lost. However, such a mixed product of measures inspires a compactification of the quotient space of orbits of product measures, which is structurally slightly different from the one introduced in citeMV14. The orbits of the product of independent occupation measures are embedded in such a compactfication and a strong large deviation principle for these objects enables us to prove the desired asymptotic localization properties of the joint behavior of two independent paths under the Gibbs transformation. As a second application, we study the spatially smoothened parabolic Anderson model in $R^d$ with white noise potential and provide a direct computation of the annealed Lyapunov exponents of the smoothened solutions when the smoothing parameter goes to $0$.
- ItemDisruptive events in high-density cellular networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Keeler, Paul; Jahnel, Benedikt; Maye, Oliver; Aschenbach, Daniel; Brzozowski, MarcinStochastic geometry models are used to study wireless networks, particularly cellular phone networks, but most of the research focuses on the typical user, often ignoring atypical events, which can be highly disruptive and of interest to network operators. We examine atypical events when a unexpected large proportion of users are disconnected or connected by proposing a hybrid approach based on ray launching simulation and point process theory. This work is motivated by recent results [12] using large deviations theory applied to the signal-to-interference ratio. This theory provides a tool for the stochastic analysis of atypical but disruptive events, particularly when the density of transmitters is high. For a section of a European city, we introduce a new stochastic model of a single network cell that uses ray launching data generated with the open source RaLaNS package, giving deterministic path loss values. We collect statistics on the fraction of (dis)connected users in the uplink, and observe that the probability of an unexpected large proportion of disconnected users decreases exponentially when the transmitter density increases. This observation implies that denser networks become more stable in the sense that the probability of the fraction of (dis)connected users deviating from its mean, is exponentially small. We also empirically obtain and illustrate the density of users for network configurations in the disruptive event, which highlights the fact that such bottleneck behaviour not only stems from too many users at the cell boundary, but also from the near-far effect of many users in the immediate vicinity of the base station. We discuss the implications of these findings and outline possible future research directions.
- ItemLarge deviations of specific empirical fluxes of independent Markov chains, with implications for Macroscopic Fluctuation Theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Renger, D.R. MichielWe consider a system of independent particles on a finite state space, and prove a dynamic large-deviation principle for the empirical measure-empirical flux pair, taking the specific fluxes rather than net fluxes into account. We prove the large deviations under deterministic initial conditions, and under random initial conditions satisfying a large-deviation principle. We then show how to use this result to generalise a number of principles from Macroscopic Fluctuation Theory to the finite-space setting.
- ItemMean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bolthausen, Erwin; König, Wolfgang; Mukherjee, ChiranjibWe consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the ``mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]
- ItemRandom walk on random walks: Low densities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Blondel, Oriane; Hilário, Marcelo R.; Santos, Renato dos; Sidoravicius, Vladas; Teixeira, AugustoWe consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or non-lazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.
- ItemRandom walk on random walks: Higher dimensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Blondel, Oriane; Hilário, Marcelo R.; Santos, Renato Soares dos; Sidoravicius, Vladas; Teixeira, AugustoWe study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].