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25
Universality of the asymptotics of the onesided exit problem for integrated processes
, 2009
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Scaling limits of (1+1)dimensional pinning models with Laplacian interaction. Ann. Probab
, 2008
"... We consider a random field ϕ:{1,...,N} → R with Laplacian interaction of the form ∑ V (∆ϕi), where ∆ is the discrete Laplacian i and the potential V (·) is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward ε ≥ 0 each time it touches the xaxis, that ..."
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Cited by 13 (2 self)
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We consider a random field ϕ:{1,...,N} → R with Laplacian interaction of the form ∑ V (∆ϕi), where ∆ is the discrete Laplacian i and the potential V (·) is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward ε ≥ 0 each time it touches the xaxis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime (ε < εc) and a localized one (ε> εc), where 0 < εc < ∞. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show, in particular, that in the delocalized regime the field wanders away from the defect line at a typical distance N 3/2, while in the localized regime the distance is just O((log N) 2). A subtle scenario shows up in the critical regime (ε = εc), where the field, suitably rescaled, converges in distribution
Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension
, 2009
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Persistence probabilities and exponents
, 1203
"... Abstract This article deals with the asymptotic behavior as t → +∞ of the survival function P[T > t], where T is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behavior is of the type P[T > t] = t −θ +o(1) ..."
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Cited by 8 (5 self)
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Abstract This article deals with the asymptotic behavior as t → +∞ of the survival function P[T > t], where T is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behavior is of the type P[T > t] = t −θ +o(1) for a known or unknown positive parameter θ which is called the persistence exponent. The problem is well understood for random walks or Lévy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.
On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data
, 2007
"... Abstract. Consider an inviscid Burgers equation whose initial data is a Lévy α−stable process Z with α> 1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. ..."
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Cited by 7 (4 self)
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Abstract. Consider an inviscid Burgers equation whose initial data is a Lévy α−stable process Z with α> 1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. This gives a negative answer to a conjecture of Janicki and Woyczynski [13]. Along the way, we contradict a recent conjecture of Z. Shi about the lower tails of integrated stable processes.
DEPINNING OF A POLYMER IN A MULTIINTERFACE MEDIUM
, 901
"... Abstract. In this paper we consider a model which describes a polymer chain interacting with an infinity of equispaced linear interfaces. The distance between two consecutive interfaces is denoted by T = TN and is allowed to grow with the size N of the polymer. When the polymer receives a positive ..."
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Cited by 4 (2 self)
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Abstract. In this paper we consider a model which describes a polymer chain interacting with an infinity of equispaced linear interfaces. The distance between two consecutive interfaces is denoted by T = TN and is allowed to grow with the size N of the polymer. When the polymer receives a positive reward for touching the interfaces, its asymptotic behavior has been derived in [3], showing that a transition occurs when TN ≈ log N. In the present paper, we deal with the so–called depinning case, i.e., the polymer is repelled rather than attracted by the interfaces. Using techniques from renewal theory, we determine the scaling behavior of the model for large N as a function of {TN}N, showing that two transitions occur, when TN ≈ N 1/3 and when TN ≈ √ N respectively. 1. Introduction and
POSITIVITY OF INTEGRATED RANDOM WALKS
"... Abstract. Take a centered random walk Sn and consider the sequence of its partial sums An: = ∑n i=1 Si. Suppose S1 is in the domain of normal attraction of an αstable law with 1 < α ≤ 2. Assuming that S1 is either rightexponential (that is P(S1> xS1> 0) = e−ax for some a> 0 and all x ..."
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Cited by 3 (0 self)
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Abstract. Take a centered random walk Sn and consider the sequence of its partial sums An: = ∑n i=1 Si. Suppose S1 is in the domain of normal attraction of an αstable law with 1 < α ≤ 2. Assuming that S1 is either rightexponential (that is P(S1> xS1> 0) = e−ax for some a> 0 and all x> 0) or rightcontinuous (skip free), we prove that
The scaling limits of the non critical strip wetting model EURANDOM PREPRINT SERIES The scaling limits of the non critical strip wetting model The scaling limits of the non critical strip wetting model
"... Abstract The strip wetting model is defined by giving a (continuous space) one dimensionnal random walk S a reward β each time it hits the strip R + × [0, a] (where a is a given positive parameter), which plays the role of a defect line. We show that this model exhibits a phase transition between a ..."
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Abstract The strip wetting model is defined by giving a (continuous space) one dimensionnal random walk S a reward β each time it hits the strip R + × [0, a] (where a is a given positive parameter), which plays the role of a defect line. We show that this model exhibits a phase transition between a delocalized regime (β < β a c ) and a localized one (β > β a c ), where the critical point β a c > 0 depends on S and on a. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. Our approach is based on Markov renewal theory.
FLUCTUATIONS FOR A CONSERVATIVE INTERFACE MODEL ON A
, 711
"... Abstract. We consider an effective interface model on a hard wall in (1+1) dimensions, with conservation of the area between the interface and the wall. We prove that the equilibrium fluctuations of the height variable converge in law to the solution of a SPDE with reflection and conservation of the ..."
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Abstract. We consider an effective interface model on a hard wall in (1+1) dimensions, with conservation of the area between the interface and the wall. We prove that the equilibrium fluctuations of the height variable converge in law to the solution of a SPDE with reflection and conservation of the space average. The proof is based on recent results obtained with L. Ambrosio and G. Savaré on stability properties of Markov processes with logconcave invariant measures. 2000 Mathematics Subject Classification: 60K35; 60H15; 82B05 1.