arXiv subjects

Stefan Grosskinsky

Publications and source records attributed to Stefan Grosskinsky.

At least 19 recordsLinked to original sources

arXiv2026

Condensation in stochastic lattice gases with size-dependent stationary weights

We consider stochastic lattice gases with stationary product weights and a polynomial perturbation vanishing with the system size that leads to condensation. If the density of particles exceeds a critical value the system phase separates into a bulk with homogeneous distribution of particles and a condensed phase. Depending on parameter values, the latter consists of a single macroscopic cluster or a diverging number of independent clusters on a smaller scale. We establish the condensation transition via the equivalence of ensembles and the main novelty is a derivation of the cluster size distribution using size-biased sampling, generalizing previous work on zero-range and inclusion processes. Simulations of zero-range processes illustrate our theoretical results on the condensate scale and size distribution.

math.PR↗

arXiv2024

Tagged particles and size-biased dynamics in mean-field interacting particle systems

We establish a connection between tagged particles and size-biased empirical processes in interacting particle systems, in analogy to classical results on the propagation of chaos. In a mean-field scaling limit, the evolution of the occupation number on the tagged particle site converges to a time-inhomogeneous Markov process with non-linear master equation given by the law of large numbers of size-biased empirical measures. The latter are important in recent efforts to understand the dynamics of condensation in interacting particle systems.

math.PR↗

arXiv2024

Condensation in zero-range processes with a fast rate

We introduce a simple zero-range process with constant rates and one fast rate for a particular occupation number, which diverges with the system size. Surprisingly, this minor modification induces a condensation transition in the thermodynamic limit, where the structure of the condensed phase depends on the scaling of the fast rate. We study this transition and its dependence on system parameters in detail on a rigorous level using size-biased sampling. This approach generalizes to any particle system with stationary product measures, and the techniques used in this paper provide a foundation for a more systematic understanding of condensing models with a non-trivial condensed phase.

math.PR↗

arXiv2024

Tails of explosive birth processes and applications to non-linear P ólya urns

We derive a simple expression for the tail-asymptotics of an explosive birth process at a fixed observation time conditioned on non-explosion. Using the well-established exponential embedding, we apply this result to compute the tail distribution of the number of balls of a losing colour in generalized P ólya urn models with super-linear feedback, which are known to exhibit a strong monopoly for the winning colour where losers only win a finite amount. Previous results in this direction were restricted to two colours with the same feedback, which we extend to an arbitrary finite number of colours with individual feedback mechanisms. As an apparent paradox, losing colours with weak feedback are more likely to win in many steps than those with strong feedback. Our approach also allows to characterize the correlations of several losing colours, which provides new insight in the distribution of other related quantities like the total number of balls for losing colours or the time to monopoly. In order to give a complete picture, we also consider sub-linear feedback and discuss asymptotics for a diverging number of colours.

math.PR↗

arXiv2024

Size-biased diffusion limits and the inclusion process

We consider the inclusion process on the complete graph with vanishing diffusivity, which leads to condensation of particles in the thermodynamic limit. Describing particle configurations in terms of size-biased and appropriately scaled empirical measures of mass distribution, we establish convergence in law of the inclusion process to a measure-valued Markov process on the space of probability measures. In the case where the diffusivity vanishes like the inverse of the system size, the derived scaling limit is equivalent to the well known Poisson-Dirichlet diffusion, offering an alternative viewpoint on these well-established dynamics. Moreover, our approach covers all scaling regimes of the system parameters and yields a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. We also discuss in detail connections to known results on related Fleming-Viot processes.

math.PR↗

arXiv2024

Wages and Capital returns in a generalized Pólya urn

It is a widely observed phenomenon that wealth is distributed significantly more unequal than wages. In this paper we study this phenomenon using a new extension of Pólyas urn, modelling wealth growth through wages and capital returns. We focus in particular on the role of increasing return rates on capital, which have been identified as a main driver of inequality, and labor share, the second main parameter of our model. We fit the parameters from real-world data in Germany, so that simulation results reproduce the empirical wealth distribution and recent dynamics in Germany quite accurately, and are essentially independent from initial conditions. Our model is simple enough to allow for a detailed mathematical analysis and provides interesting predictions for future developments and on the importance of wages and capital returns for wealth aggregation. We also provide an extensive discussion of the robustness of our results and the plausibility of the main assumptions used in our model.

math.PR↗

arXiv2023

Asymptotics of generalized Pólya urns with non-linear feedback

Generalized Pólya urns with non-linear feedback are an established probabilistic model to describe the dynamics of growth processes with reinforcement, a generic example being competition of agents in evolving markets. It is well known which conditions on the feedback mechanism lead to monopoly where a single agent achieves full market share, and various further results for particular feedback mechanisms have been derived from different perspectives. In this paper we provide a comprehensive account of the possible asymptotic behaviour for a large general class of feedback, and describe in detail how monopolies emerge in a transition from sub-linear to super-linear feedback via hierarchical states close to linearity. We further distinguish super- and sub-exponential feedback, which show conceptually interesting differences to understand the monopoly case, and study robustness of the asymptotics with respect to initial conditions, heterogeneities and small changes of the feedback mechanisms. Finally, we derive a scaling limit for the full time evolution of market shares in the limit of diverging initial market size, including the description of typical fluctuations and extending previous results in the context of stochastic approximation.

math.PR↗

arXiv2022

Poisson-Dirichlet asymptotics in condensing particle systems

We study measures on random partitions, arising from condensing stochastic particle systems with stationary product distributions. We provide fairly general conditions on the stationary weights, which lead to Poisson-Dirichlet statistics of the condensed phase in the thermodynamic limit. The Poisson-Dirichlet distribution is known to be the unique reversible measure of split-merge dynamics for random partitions, which we use to characterize the limit law. We also establish concentration results for the macroscopic phase, using size-biased sampling techniques and the equivalence of ensembles to characterize the bulk distribution of the system.

math.PR↗

arXiv2021

A Study of UK Household Wealth through Empirical Analysis and a Non-linear Kesten Process

We study the wealth distribution of UK households through a detailed analysis of data from wealth surveys and rich lists, and propose a non-linear Kesten process to model the dynamics of household wealth. The main features of our model are that we focus on wealth growth and disregard exchange, and that the rate of return on wealth is increasing with wealth. The linear case with wealth-independent return rate has been well studied, leading to a log-normal wealth distribution in the long time limit which is essentially independent of initial conditions. We find through theoretical analysis and simulations that the non-linearity in our model leads to more realistic power-law tails, and can explain an apparent two-tailed structure in the empirical wealth distribution of the UK and other countries. Other realistic features of our model include an increase in inequality over time, and a stronger dependence on initial conditions compared to linear models.

econ.TH↗

arXiv2021

Limit theorems for cloning algorithms

Large deviations for additive path functionals of stochastic processes have attracted significant research interest, in particular in the context of stochastic particle systems and statistical physics. Efficient numerical `cloning' algorithms have been developed to estimate the scaled cumulant generating function, based on importance sampling via cloning of rare event trajectories. So far, attempts to study the convergence properties of these algorithms in continuous time have only led to partial results for particular cases. Adapting previous results from the literature of particle filters and sequential Monte Carlo methods, we establish a first comprehensive and fully rigorous approach to bound systematic and random errors of cloning algorithms in continuous time. To this end we develop a method to compare different algorithms for particular classes of observables, based on the martingale characterization of stochastic processes. Our results apply to a large class of jump processes on compact state space, and do not involve any time discretization in contrast to previous approaches. This provides a robust and rigorous framework that can also be used to evaluate and improve the efficiency of algorithms.

math.PR↗

arXiv2019

Structure of the condensed phase in the inclusion process

We establish a complete picture of condensation in the inclusion process in the thermodynamic limit with vanishing diffusion, covering all scaling regimes of the diffusion parameter and including large deviation results for the maximum occupation number. We make use of size-biased sampling to study the structure of the condensed phase, which can extend over more than one lattice site and exhibit an interesting hierarchical structure characterized by the Poisson-Dirichlet distribution. While this approach is established in other areas including population genetics or random permutations, we show that it also provides a powerful tool to analyse homogeneous condensation in stochastic particle systems with stationary product distributions. We discuss the main mechanisms beyond inclusion processes that lead to the interesting structure of the condensed phase, and the connection to other generic particle systems. Our results are exact, and we present Monte-Carlo simulation data and recursive numerics for partition functions to illustrate the main points.

math.PR↗

arXiv2019

Rare event simulation for stochastic dynamics in continuous time

Large deviations for additive path functionals of stochastic dynamics and related numerical approaches have attracted significant recent research interest. We focus on the question of convergence properties for cloning algorithms in continuous time, and establish connections to the literature of particle filters and sequential Monte Carlo methods. This enables us to derive rigorous convergence bounds for cloning algorithms which we report in this paper, with details of proofs given in a further publication. The tilted generator characterizing the large deviation rate function can be associated to non-linear processes which give rise to several representations of the dynamics and additional freedom for associated numerical approximations. We discuss these choices in detail, and combine insights from the filtering literature and cloning algorithms to compare different approaches and improve efficiency.

cond-mat.stat-mech↗

arXiv2018

Coarsening dynamics in a two-species zero-range process

We consider a zero-range process with two species of interacting particles. The steady state phase diagram of this model shows a variety of condensate phases in which a single site contains a finite fraction of all the particles in the system. Starting from a homogeneous initial distribution, we study the coarsening dynamics in each of these condensate phases, which is expected to follow a scaling law. Random walk arguments are used to predict the coarsening exponents in each condensate phase. They are shown to depend on the form of the hop rates and on the symmetry of the hopping dynamics. The analytic predictions are found to be in good agreement with the results of Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2018

Dynamical origin of spontaneous symmetry breaking in a field-driven nonequilibrium system

A one-dimensional driven two-species model with parallel sublattice update and open boundaries is considered. Although the microscopic many-body dynamics is symmetric with respect to the two species and interactions are short-ranged, there is a region in parameter space with broken symmetry in the steady state. The sublattice update is deterministic in the bulk and allows for a detailed analysis of the relaxation dynamics, so that symmetry breaking can be shown to be the result of an amplification mechanism of fluctuations. In contrast to previously considered models, this leads to a proof for spontaneous symmetry breaking which is valid throughout the whole region in parameter space with a symmetry broken steady state.

cond-mat.stat-mech↗

arXiv2018

Current large deviations for partially asymmetric particle systems on a ring

We study large deviations for the current of one-dimensional stochastic particle systems with periodic boundary conditions. Following a recent approach based on an earlier result by Jensen and Varadhan, we compare several candidates for atypical currents to travelling wave density profiles, which correspond to non-entropic weak solutions of the hyperbolic scaling limit of the process. We generalize previous results to partially asymmetric systems and systems with convex as well as concave current-density relations, including zero-range and inclusion processes. We provide predictions for the large deviation rate function covering the full range of current fluctuations using heuristic arguments, and support them by simulation results using cloning algorithms wherever they are computationally accessible. For partially asymmetric zero-range processes we identify an interesting dynamic phase transition between different strategies for atypical currents, which is of a generic nature and expected to apply to a large class of particle systems on a ring.

cond-mat.stat-mech↗

arXiv2018

Derivation of mean-field equations for stochastic particle systems

We study stochastic particle systems on a complete graph and derive effective mean-field rate equations in the limit of diverging system size, which are also known from cluster aggregation models. We establish the propagation of chaos under generic growth conditions on particle jump rates, and the limit provides a master equation for the single site dynamics of the particle system, which is a non-linear birth death chain. Conservation of mass in the particle system leads to conservation of the first moment for the limit dynamics, and to non-uniqueness of stationary distributions. Our findings are consistent with recent results on exchange driven growth, and provide a connection between the well studied phenomena of gelation and condensation.

math.PR↗

arXiv2017

Coarsening dynamics in condensing zero-range processes and size-biased birth death chains

Zero-range processes with decreasing jump rates are well known to exhibit a condensation transition under certain conditions on the jump rates, and the dynamics of this transition continues to be a subject of current research interest. Starting from homogeneous initial conditions, the time evolution of the condensed phase exhibits an interesting coarsening phenomenon of mass transport between cluster sites characterized by a power law. We revisit the approach in [C. Godreche, J. Phys. A: Math. Gen., 36(23) 6313 (2003)] to derive effective single site dynamics which form a non-linear birth death chain describing the coarsening behaviour. We extend these results to a larger class of parameter values, and introduce a size-biased version of the single site process, which provides an effective tool to analyze the dynamics of the condensed phase without finite size effects and is the main novelty of this paper. Our results are based on a few heuristic assumptions and exact computations, and are corroborated by detailed simulation data.

cond-mat.stat-mech↗

arXiv2017

Lower current large deviations for zero-range processes on a ring

We study lower large deviations for the current of totally asymmetric zero-range processes on a ring with concave current-density relation. We use an approach by Jensen and Varadhan which has previously been applied to exclusion processes, to realize current fluctuations by travelling wave density profiles corresponding to non-entropic weak solutions of the hyperbolic scaling limit of the process. We further establish a dynamic transition, where large deviations of the current below a certain value are no longer typically attained by non-entropic weak solutions, but by condensed profiles, where a non-zero fraction of all the particles accumulates on a single fixed lattice site. This leads to a general characterization of the rate function, which is illustrated by providing detailed results for four generic examples of jump rates, including constant rates, decreasing rates, unbounded sublinear rates and asymptotically linear rates. Our results on the dynamic transition are supported by numerical simulations using a cloning algorithm.

cond-mat.stat-mech↗