Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

Ioffe, Dmitry; Velenik, Yvan; Wachtel, Vitali

doi:10.1007/s00440-016-0751-z

Cited by 23 publications

(29 citation statements)

References 23 publications

(40 reference statements)

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“…as L → ∞ converges weakly to the stationary Ferrari-Spohn diffusion, namely the reversible diffusion process on R + with potential given by the logarithm of the Airy function, which was first introduced in [11]. On the other hand, for fixed n, and for λ = 1, the ensemble of random lines described by (1.1) has been analysed in [16], where it is shown that the vector of rescaled trajectories…”

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confidence: 99%

Confinement of Brownian polymers under geometric area tilts

Caputo¹,

Ioffe²,

Wachtel³

2019

Electron. J. Probab.

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confidence: 99%

Confinement of Brownian polymers under geometric area tilts

Caputo¹,

Ioffe²,

Wachtel³

2019

Electron. J. Probab.

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“…As a result of this progress the semicircular case may have lost some of its initial motivation. However, Brownian excursion, conditioned to stay away from a moving wall, continues to attract attention, and it has been recently generalized in several directions [9][10][11][12][13]. 2 Already in their original paper [3] the authors noticed that finer details of their model differ from those of models of the KPZ universality class.…”

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confidence: 99%

Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions

Smith,

Meerson

2018

Preprint

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We study a Brownian excursion on the time interval |t| ≤ T , conditioned to stay above a moving wall x0 (t) such that x0 (−T ) = x0 (T ) = 0, and x0 (|t| < T ) > 0. For a whole class of moving walls, typical fluctuations of the conditioned Brownian excursion are described by the Ferrari-Spohn (FS) distribution and exhibit the Kardar-Parisi-Zhang (KPZ) dynamic scaling exponents 1/3 and 2/3. Here we use the optimal fluctuation method (OFM) to study atypical fluctuations, which turn out to be quite different. The OFM provides their simple description in terms of optimal paths, or rays, of the Brownian motion. We predict two singularities of the large deviation function, which can be interpreted as dynamical phase transitions, and they are typically of third order. Transitions of a fractional order can also appear depending on the behavior of x0 (t) in a close vicinity of t = ±T . Although the OFM does not describe typical fluctuations, it faithfully reproduces the near tail of the FS distribution and therefore captures the KPZ scaling. If the wall function x0 (t) is not parabolic near its maximum, typical fluctuations (which we probe in the near tail) exhibit a more general scaling behavior with a continuous one-parameter family of scaling exponents.

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“…Another important difference is that in the SOS model [10] one never sees the top monolayer detached from the rest of them, as in the model we consider. Nevertheless, we believe that in our model with k monolayers the fluctuations of their boundaries in the vicinity of the (vertical) wall are, as in the case of entropic repulsion [11], of the order of N 1/3 , and their behavior is given, after appropriate scaling, by k non-intersecting Ferrari-Spohn diffusions [23], as in [26,29]. See [28] for a review.…”

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Formation of Facets for an Effective Model of Crystal Growth

Ioffe¹,

Shlosman²

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We study an effective model of microscopic facet formation for low temperature three dimensional microscopic Wulff crystals above the droplet condensation threshold. The model we consider is a 2+1 solid on solid surface coupled with high and low density bulk Bernoulli fields. At equilibrium the surface stays flat. Imposing a canonical constraint on excess number of particles forces the surface to "grow" through the sequence of spontaneous creations of macroscopic size monolayers. We prove that at all sufficiently low temperatures, as the excess particle constraint is tuned, the model undergoes an infinite sequence of first order transitions, which traces an infinite sequence of first order transitions in the underlying variational problem. Away from transition values of canonical constraint we prove sharp concentration results for the rescaled level lines around solutions of the limiting variational problem.

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Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

Cited by 23 publications

References 23 publications

Confinement of Brownian polymers under geometric area tilts

Confinement of Brownian polymers under geometric area tilts

Geometrical optics of constrained Brownian excursion: from the KPZ scaling to dynamical phase transitions

Formation of Facets for an Effective Model of Crystal Growth

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