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- ItemQuenched large deviations for simple random walks on percolation clusters including long-range correlations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Berger, Noam; Mukherjee, Chiranjib; Okamura, KazukiWe prove a quenched large deviation principle (LDP)for a simple random walk on a supercritical percolation cluster (SRWPC) on the lattice.The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and its vacant set and the level sets of the Gaussian free field. Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures if the environment Markov chain in the non-elliptic case of SRWPC. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main approach of our proofs are based on exploiting coercivity properties of the relative entropy in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the percolation cluster under supercriticality.
- ItemLarge deviations for the capacity in dynamic spatial relay networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Hirsch, Christian; Jahnel, BenediktWe derive a large deviation principle for the space-time evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties.
- ItemFrom large deviations to Wasserstein gradient flows in multiple dimensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Erbar, Matthias; Maas, Jan; Renger, D.R. MichielWe study the large deviation rate functional for the empirical measure of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of -convergence) to the JordanKinderlehrerOtto functional arising in the Wasserstein gradient flow structure of the FokkerPlanck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.
- ItemSpace-time large deviations in capacity-constrained relay networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hirsch, Christian; Jahnel, Benedikt; Patterson, RobertWe consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime.
- ItemDynamical phase transitions for flows on finite graphs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Gabrielli, Davide; Renger, D. R. MichielWe study the time-averaged flow in a model of particles that randomly hop on a finite directed graph. In the limit as the number of particles and the time window go to infinity but the graph remains finite, the large-deviation rate functional of the average flow is given by a variational formulation involving paths of the density and flow. We give sufficient conditions under which the large deviations of a given time averaged flow is determined by paths that are constant in time. We then consider a class of models on a discrete ring for which it is possible to show that a better strategy is obtained producing a time-dependent path. This phenomenon, called a dynamical phase transition, is known to occur for some particle systems in the hydrodynamic scaling limit, which is thus extended to the setting of a finite graph.
- ItemLarge deviations for Markov jump processes with uniformly diminishing rates(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Agazzi, Andrea; Andreis, Luisa; Patterson, Robert I. A.; Renger, D. R. MichielWe prove a large-deviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further show that our assumptions on the decay of the jump rates are optimal. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of Mass action kinetics.
- ItemLarge deviations of reaction fluxes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Patterson, Robert I.A.; Renger, D.R. MichielWe study a system of interacting particles that randomly react to form new particles. The reaction flux is the rescaled number of reactions that take place in a time interval. We prove a dynamic large-deviation principle for the reaction fluxes under general assumptions that include mass-action kinetics. This result immediately implies the dynamic large deviations for the empirical concentration.
- ItemDynamical large deviations of countable reaction networks under a weak reversibility condition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Patterson, Robert I.A.; Renger, D.R. MichielA dynamic large deviations principle for a countable reaction network including coagulation-fragmentation models is proved. The rate function is represented as the infimal cost of the reaction fluxes and a minimiser for this variational problem is shown to exist. A weak reversibility condition is used to control the boundary behaviour and to guarantee a representation for the optimal fluxes via a Lagrange multiplier that can be used to construct the changes of measure used in standard tilting arguments. Reflecting the pure jump nature of the approximating processes, their paths are treated as elements of a BV function space.
- ItemGradient and Generic systems in the space of fluxes, applied to reacting particle systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Renger, D.R. MichielIn a previous work we devised a framework to derive generalised gradient systems for an evolution equation from the large deviations of an underlying microscopic system, in the spirit of the Onsager-Machlup relations. Of particular interest is the case where the microscopic system consists of random particles, and the macroscopic quantity is the empirical measure or concentration. In this work we take the particle flux as the macroscopic quantity, which is related to the concentration via a continuity equation. By a similar argument the large deviations can induce a generalised gradient or Generic system in the space of fluxes. In a general setting we study how flux gradient or generic systems are related to gradient systems of concentrations. The arguments are explained by the example of reacting particle systems, which is later expanded to include spatial diffusion as well.