Conditioned random walks and interaction-driven condensation

Abstract

We consider a discrete-time continuous-space random walk under the constraints that the number of returns to the origin (local time) and the total area under the walk are fixed. We first compute the joint probability of an excursion having area $a$ and returning to the origin for the first time after time $\tau$. We then show how condensation occurs when the total area constraint is increased: an excursion containing a finite fraction of the area emerges. Finally we show how the phenomena generalises previously studied cases of condensation induced by several constraints and how it is related to interaction-driven condensation which allows us to explain the phenomenon in the framework of large deviation theory.

28 pages, 6 figures, invited paper for Special Issue of J. Phys. A "Emerging talents"

References in corpus (31)

• The large deviation approach to statistical mechanics
• Non equilibrium steady states: fluctuations and large deviations of the density and of the current
• Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Nonequilibrium Markov processes conditioned on large deviations
• Macroscopic current fluctuations in stochastic lattice gases
• Distribution of current in non-equilibrium diffusive systems and phase transitions
• Nonequilibrium microcanonical and canonical ensembles and their equivalence
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Condensation in the zero range process: stationary and dynamical properties
• Nonequilibrium Phase Transitions in Models of Aggregation, Adsorption, and Dissociation
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• The Nature of the Condensate in Mass Transport Models
• Canonical Analysis of Condensation in Factorised Steady State
• Interaction driven real-space condensation
• On the area under a continuous time Brownian motion till its first-passage time
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Constraint driven condensation in large fluctuations of linear statistics
• Evolutionary games of condensates in coupled birth-death processes
• Conditional Distribution of Heavy Tailed Random Variables on Large Deviations of their Sum
• Zero Range Condensation at Criticality
• Condensation in Temporally Correlated Zero-Range Dynamics
• Tuning the shape of the condensate in spontaneous symmetry breaking
• Inequivalence of nonequilibrium path ensembles: the example of stochastic bridges
• Condensation of Fluctuations in and out of Equilibrium
• Condensation transition in joint large deviations of linear statistics
• The grand canonical catastrophe as an instance of condensation of fluctuations
• Mass condensation in one dimension with pair-factorized steady states
• Tails of probability density for sums of random independent variables
• Numerical survey of the tunable condensate shape and scaling laws in pair-factorized steady states
• Maintenance of order in a moving strong condensate

Cited by in corpus (9)

• A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
• Condensation transition in the late-time position of a Run-and-Tumble particle
• Localization transition in the Discrete Non-Linear Schrödinger Equation: ensembles inequivalence and negative temperatures
• Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process
• A comparison of dynamical fluctuations of biased diffusion and run-and-tumble dynamics in one dimension
• Participation ratio for constraint-driven condensation with superextensive mass
• Probability distributions with singularities
• A sluggish random walk with subdiffusive spread
• First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension