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19
Limit laws for transient random walks in random environment on Z
, 2007
"... We consider transient random walks in random environment on Z with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description o ..."
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Cited by 29 (8 self)
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We consider transient random walks in random environment on Z with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.
Universality and extremal aging for dynamics of spin glasses on subexponential time scales
 Comm. Pure Appl. Math
, 2012
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AGING IN REVERSIBLE DYNAMICS OF DISORDERED SYSTEMS. II. emergence of the arcsine . . .
, 2010
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Convergence of clock process in random environments and aging in Bouchaud’s asymmetric trap model on the complete graph
, 2012
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Phase transition for the speed of the biased random walk on the supercritical percolation cluster
"... Abstract. We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on Z d. That is, for each d ≥ 2, and for any supercritical parameter p> pc, we prove the existence of a critical strength for the bias, such that, below this value ..."
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Abstract. We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on Z d. That is, for each d ≥ 2, and for any supercritical parameter p> pc, we prove the existence of a critical strength for the bias, such that, below this value, the speed is positive, and, above the value, it is zero. We identify the value of the critical bias explicitly, and, in the subballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk. A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially onedimensional; we prove such a dynamic renormalization statement in a much stronger form than was previously known. 1.
Ageing in the parabolic Anderson model
"... The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on timeconstant, independent and identically distributed potentials with polynomial tails at infinity. ..."
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Cited by 6 (2 self)
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The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on timeconstant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the longterm temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.
AGING IN REVERSIBLE DYNAMICS OF DISORDERED SYSTEMS. I. emergence of the arcsine . . .
, 2010
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Convergence to extremal processes in random environments and extremal ageing in SK models
, 2012
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SCALING LIMIT AND AGING FOR DIRECTED TRAP MODELS
, 2008
"... We consider onedimensional directed trap models and suppose that the trapping times are heavytailed. We obtain the inverse of a stable subordinator as scaling limit and prove aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality desc ..."
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Cited by 4 (0 self)
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We consider onedimensional directed trap models and suppose that the trapping times are heavytailed. We obtain the inverse of a stable subordinator as scaling limit and prove aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and Čern´y for a large class of graphs.