1. A limit theorem for the survival probability of a simple random walk among power-law renewal traps
- Author
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François Simenhaus, Julien Poisat, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), ANR-16-CE93-0003,MALIN,Marches aléatoires en interaction(2016), ANR-15-CE40-0020,LSD,Modèles stochastiques en grande dimension pour la physique statistique hors équilibre(2015), and ANR-17-CE40-0032,SWiWS,Gruyère et Saucisse de Wiener(2017)
- Subjects
Statistics and Probability ,polymers in random environments ,Logarithm ,01 natural sciences ,Power law ,010104 statistics & probability ,Poisson point process ,FOS: Mathematics ,Random walks in random obstacles ,Renewal theory ,Statistical physics ,0101 mathematics ,survival probability ,60K37 60K35 ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,FKG inequalities ,Ray–Knight theorems ,010102 general mathematics ,Probability (math.PR) ,Hitting time ,parabolic Anderson model ,Partition function (mathematics) ,16. Peace & justice ,Random walk ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Scaling limit ,60K37 ,60K35 ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
We consider a one-dimensional simple random walk surviving among a field of static soft traps : each time it meets a trap the walk is killed with probability 1--e --$\beta$ , where $\beta$ is a positive and fixed parameter. The positions of the traps are sampled independently from the walk and according to a renewal process. The increments between consecutive traps, or gaps, are assumed to have a power-law decaying tail with exponent $\gamma$ > 0. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability as time goes to infinity. The normalization exponent is $\gamma$/($\gamma$ + 2), while the limiting law writes as a variational formula with both universal and non-universal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a function of the parameter $\beta$ that we call asymptotic cost of crossing per trap and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap. This model may also be seen as a (1 + 1)-directed polymer among many repulsive interfaces, in which case $\beta$ corresponds to the strength of repulsion, the survival probability to the partition function and its logarithm to the finite-volume free energy. Along the way we prove a stochastic monotonicity property for the hitting time of the killed random walk with respect to the non-killed one, that could be of interest in other contexts, see Proposition 3.5., Comment: Version 2: shorter proof of Proposition 3.5
- Published
- 2018
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