The Brownian web: Characterization and convergence

“…Nr, The one dimensional trap model (hereafter denoted as 1DTM) has been the focus of renewed attention, both in the mathematical community [1,2], and also using a physicist approach [3,4]. This model was proposed in the 70's to describe the properties of one dimensional disordered conductors [5,6].…”

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confidence: 99%

“…Once that particle has escaped the trap, it chooses one of the two nearest neighbouring sites, with probability q − for the left one and q + = 1 − q − for the right one. Two particular cases have already been studied in details in the literature, namely the unbiased case q + = 1/2 [3,1,2], and the fully directed case q + = 1 [12,6,13].…”

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confidence: 99%

Linear and nonlinear response in the aging regime of the one-dimensional trap model

2003

Phys. Rev. E

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We investigate the behaviour of the response function in the one dimensional trap model using scaling arguments that we confirm by numerical simulations. We study the average position of the random walk at time tw + t given that a small bias h is applied at time tw. Several scaling regimes are found, depending on the relative values of t, tw and h. Comparison with the diffusive motion in the absence of bias allows us to show that the fluctuation dissipation relation is, in this case, valid even in the aging regime.PACS numbers: 75.10. Nr, The one dimensional trap model (hereafter denoted as 1DTM) has been the focus of renewed attention, both in the mathematical community [1,2], and also using a physicist approach [3,4]. This model was proposed in the 70's to describe the properties of one dimensional disordered conductors [5,6]. Very recently, the direct relevance of this model for the dynamics of DNA 'bubbles' under torsion was emphasized [7]. Although the "annealed" version of the model, in which a new trapping time is randomly choosen at each step, is now well documented [8][9][10], the full analysis of the quenched model considered here remains challenging. It was shown in [3,4] that, besides interesting dynamical localization properties, the 1DTM exhibits different time scalings, depending on which correlation function is considered. To be more specific, the probability of not moving between t w and t w + t -called Π(t w + t, t w ) in [3] -and the probability of occupying the same site at t w and t w + t -called C(t w + t, t w ) -exhibit different scalings: Π scales as t/t ν w with ν < 1, whereas C behaves as t/t w . Then a natural question arises in this context: what is the relevant time scale that governs the response function of the particle to an external bias? This question is interesting in the context of the physical applications mentioned above, and also in the context of the out-of-equilibrium Fluctuation Dissipation Theorem that was much discussed recently [11].Let us first briefly recall the definition of the 1DTM. Consider a one dimensional lattice, and associate to each site i a quenched random variable E i > 0, the energy barrier, chosen from a distribution ρ(E). One follows the evolution of a particle driven by a thermal noise at temperature T on the lattice. The particle has to overcome the energy barrier E i in order to leave site i and reach one neighbour. This naturally leads to an escape rate w i of site i given by an Arrhenius law w i = Γ 0 e −Ei/T . Note that the transition rates are a coarse grained description of the underlying Langevin dynamics, which is not explicitely described in this model. The quantity Γ 0 is a microscopic frequency scale, that will be set to unity in the following. Once that particle has escaped the trap, it chooses one of the two nearest neighbouring sites, with probability q − for the left one and q + = 1 − q − for the right one. Two particular cases have already been studied in details in the literature, namely the unbiased case q + = 1/2 [3,1,2], and the ...

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“…We call this object, which is closely related to the construction in [Hut09], the Brownian fan (see Section 2.2). It also has a flavour very similar to the construction of the Brownian web [FINR04] and the Brownian net [SS08], although there does not seem to be an obvious transformation linking these objects.…”

Section: Notationsmentioning

confidence: 99%

The Brownian Fan

Hairer

Weare

2014

Comm Pure Appl Math

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We provide a mathematical study of the modified Diffusion Monte Carlo (DMC) algorithm introduced in the companion article [HW14]. DMC is a simulation technique that uses branching particle systems to represent expectations associated with FeynmanKac formulae. We provide a detailed heuristic explanation of why, in cases in which a stochastic integral appears in the Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering, and other settings), the new algorithm is expected to converge in a suitable sense to a limiting process as the time interval between branching steps goes to 0. The situation studied here stands in stark contrast to the "naïve" generalisation of the DMC algorithm which would lead to an exponential explosion of the number of particles, thus precluding the existence of any finite limiting object. Convergence is shown rigorously in the simplest possible situation of a random walk, biased by a linear potential. The resulting limiting object, which we call the "Brownian fan", is a very natural new mathematical object of independent interest.

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The Brownian web: Characterization and convergence

Cited by 131 publications

References 28 publications

Scaling Limit for a Drainage Network Model

Scaling Limit for a Drainage Network Model

Linear and nonlinear response in the aging regime of the one-dimensional trap model

The Brownian Fan

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