Winding of planar Gaussian processes

Doussal, Pierre Le; Etzioni, Yoav; Horovitz, Baruch

doi:10.1088/1742-5468/2009/07/p07012

Cited by 5 publications

(11 citation statements)

References 33 publications

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“…provided that this infinite integral is finite. This coincides with predictions in a physics paper of Le Doussal, Etzioni and Horovitz [6]. They also noticed the following simplification in this case: denoting θ(x) = arcsin r(x) (θ is well-defined, since r is now real-valued), we have…”

Section: Denoting R(x) = R ′supporting

confidence: 89%

“…However, perhaps surprisingly, the winding of Gaussian stationary processes appears to be a topic that has been largely ignored. Prior to this work, we know only of a paper by Le-Doussal, Etzioni and Horovitz [6] which provides predictions and intriguing examples regarding the nature of the fluctuations of the winding. Their interest was inspired by their research on the winding of particles in random environments [9].…”

Section: Introductionmentioning

confidence: 99%

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The winding of stationary Gaussian processes

Buckley

Feldheim

2017

Probab. Theory Relat. Fields

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This paper studies the winding of a continuously differentiable Gaussian stationary process f : R → C in the interval [0, T ]. We give formulae for the mean and the variance of this random variable. The variance is shown to always grow at least linearly with T , and conditions for it to be asymptotically linear or quadratic are given. Moreover, we show that if the covariance function together with its second derivative are in L 2 (R), then the winding obeys a central limit theorem. These results correspond to similar results for zeroes of real-valued stationary Gaussian functions by Malevich, Cuzick, Slud and others.

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Section: Denoting R(x) = R ′supporting

confidence: 89%

Section: Introductionmentioning

confidence: 99%

The winding of stationary Gaussian processes

Buckley

Feldheim

2017

Probab. Theory Relat. Fields

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“…Recently Le Doussal [68] used the replica Bethe ansatz to obtain an exact representation for the distribution of the one-point, two-time height difference h (0, t) − h (0, 0) for the KPZ interface for a combined initial condition which is flat at x < 0 and stationary at x > 0. This problem can be generalized by considering the distribution P (H, L) of the two-point, two-time height difference H = h (L, t)−h (0, 0) with the same combined initial condition.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…It is therefore the optimal history for the combined initial condition of Ref. [68] at L ≥ 0. As a result, the λH 1 tail of P (H, L) for the combined initial condition is given by Eqs.…”

Section: Summary and Discussionmentioning

confidence: 99%

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Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface

2018

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We study the short-time distribution P(H,L,t) of the two-point two-time height difference H=h(L,t)-h(0,0) of a stationary Kardar-Parisi-Zhang interface in 1+1 dimension. Employing the optimal-fluctuation method, we develop an effective Landau theory for the second-order dynamical phase transition found previously for L=0 at a critical value H=H_{c}. We show that |H| and L play the roles of inverse temperature and external magnetic field, respectively. In particular, we find a first-order dynamical phase transition when L changes sign, at supercritical H. We also determine analytically P(H,L,t) in several limits away from the second-order transition. Typical fluctuations of H are Gaussian, but the distribution tails are highly asymmetric. The tails -lnP∼|H|^{3/2}/sqrt[t] and -lnP∼|H|^{5/2}/sqrt[t], previously found for L=0, are enhanced for L≠0. At very large |L| the whole height-difference distribution P(H,L,t) is time-independent and Gaussian in H, -lnP∼|H|^{2}/|L|, describing the probability of creating a ramplike height profile at t=0.

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Winding statistics of a Brownian particle on a ring

Kundu¹,

Comtet²,

Majumdar³

2014

J. Phys. A: Math. Theor.

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We consider a Brownian particle moving on a ring. We study the probability distributions of the total number of turns and the net number of counter-clockwise turns the particle makes till time t. Using a method based on the renewal properties of Brownian walker, we find exact analytical expressions of these distributions. This method serves as an alternative to the standard path integral techniques which are not always easily adaptable for certain observables. For large t, we show that these distributions have Gaussian scaling forms. We also compute large deviation functions associated to these distributions characterizing atypically large fluctuations. We provide numerical simulations in support of our analytical results.PACS numbers:

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Winding of planar Gaussian processes

Cited by 5 publications

References 33 publications

The winding of stationary Gaussian processes

The winding of stationary Gaussian processes

Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface

Winding statistics of a Brownian particle on a ring

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