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- ItemLarge deviations for Brownian intersection measures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) König, Wolfgang; Mukherjee, ChiranjibWe consider $p$ independent Brownian motions in $R^d$. We assume that $pgeq 2$ and $p(d-2)
- ItemBranching random walks in random environment: A survey(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) König, WolfgangWe consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of the observed prominent effect of intermittency (local concentration of the mass of the solution in small islands). We present several advances in the investigation of this effect, also related to questions inspired from biology.
- ItemA variational formula for the free energy of an interacting many-particle system(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Adams, Stefan; Collevecchio, Andrea; König, WolfgangWe consider $N$ bosons in a box in $R^d$ with volume $N/rho$ under the influence of a mutually repellent pair potential. The particle density $rhoin(0,infty)$ is kept fixed. Our main result is the identification of the limiting free energy, $f(beta,rho)$, at positive temperature $1/beta$, in terms of an explicit variational formula, for any fixed $rho$ if $beta$ is sufficiently small, and for any fixed $beta$ if $rho$ is sufficiently small. The thermodynamic equilibrium is described by the symmetrised trace of $rm e^-beta Hcal_N$, where $Hcal_N$ denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of $N$ interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. In our proof of the lower bound for the free energy, we drop all interaction involving lq infinitely longrq cycles, and their possible presence is signalled by a loss of mass of the lq finitely longrq cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the lq finitely longrq cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyse it further in future.
- ItemAn effective medium approach to the asymptotics of the statistical moments of the parabolic Anderson model and Lifshitz tails : dedicated to Peter Stollmann on the occasion of his 50th birthday(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Metzger, Bernd; Stollmann, PeterOriginally introduced in solid state physics to model amorphous materials and alloys exhibiting disorder induced metal-insulator transitions, the Anderson model $H_omega= -Delta + V_omega $ on $l^2(bZ^d)$ has become in mathematical physics as well as in probability theory a paradigmatic example for the relevance of disorder effects. Here $Delta$ is the discrete Laplacian and $V_omega = V_omega(x): x in bZ^d$ is an i.i.d. random field taking values in $bR$. A popular model in probability theory is the parabolic Anderson model (PAM), i.e. the discrete diffusion equation $partial_t u(x,t) =-H_omega u(x,t)$ on $ bZ^d times bR_+$, $u(x,0)=1$, where random sources and sinks are modelled by the Anderson Hamiltonian. A characteristic property of the solutions of (PAM) is the occurrence of intermittency peaks in the large time limit. These intermittency peaks determine the thermodynamic observables extensively studied in the probabilistic literature using path integral methods and the theory of large deviations. The rigorous study of the relation between the probabilistic approach to the parabolic Anderson model and the spectral theory of Anderson localization is at least mathematically less developed. We see our publication as a step in this direction. In particular we will prove an unified approach to the transition of the statistical moments $langle u(0,t) rangle$ and the integrated density of states from classical to quantum regime using an effective medium approach. As a by-product we will obtain a logarithmic correction in the traditional Lifshitz tail setting when $V_omega$ satisfies a fat tail condition.
- ItemOrthogonality of fluxes in general nonlinear reaction networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Renger, D.R.Michiel; Zimmer, JohannesWe 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.
- ItemAnisothermal chemical reactions: Onsager--Machlup and macroscopic fluctuation theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Renger, D. R. MichielWe study a micro and macroscopic model for chemical reactions with feedback between reactions and temperature of the solute. The first result concerns the quasipotential as the large-deviation rate of the microscopic invariant measure. The second result is an application of modern Onsager-Machlup theory to the pathwise large deviations, in case the system is in detailed balance. The third result is an application of macroscopic fluctuation theory to the reaction flux large deviations, in case the system is in complex balance.
- ItemA large-deviations principle for all the components in a sparse inhomogeneous random graph(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Andreis, Luisa; König, Wolfgang; Langhammer, Heide; Patterson, Robert I. A.We study an inhomogeneous sparse random graph, GN, on [N] = { 1,...,N } as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N → ∞ , we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size ≈ N). In doing so, we derive explicit logarithmic asymptotics for the probability that GN is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of GN. In particular, we recover the criterion for the existence of the phase transition given in [BJR07].
- 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.
- ItemMoment asymptotics for multitype branching random walks in random environment(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Gün, Onur; König, Wolfgang; Sekulovic, OzrenWe study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter 1/pij for the upper tail of the mean number of j type particles produced by an i type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.
- 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.
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