An Invariance Principle to Ferrari–Spohn Diffusions

The Airy distribution (AD) describes the probability distribution of the area under a Brownian excursion. The AD is prominent in several areas of physics, mathematics and computer science. Here we use a dilute colloidal system to directly measure, for the first time, the AD in experiment. We also show how two different techniques of theory of large deviations -the Donsker-Varadhan formalism and the optimal fluctuation method -manifest themselves in the AD. We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area. For large areas, we uncover two singularities in the large deviation function, which can be interpreted as dynamical phase transitions of third order. For small areas the position distribution coincides with the Ferrari-Spohn distribution, which has previously appeared in other problems, and we identify the reason for this coincidence.

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“…The FS distribution has also appeared in other systems [36,[52][53][54][55]. This is indicative of a universality class.…”

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confidence: 68%

Airy distribution: Experiment, large deviations, and additional statistics

Agranov

Zilber

Smith

et al. 2020

Phys. Rev. Research

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“…Let u, v ∈ N. As in [15], P u,v M,N,+ is used to denote the set of trajectories X starting at u at time M , ending at v at time N and staying positive during the time interval {M, . .…”

Section: 1mentioning

confidence: 99%

“…(6.14) Since, evidently, 15) all the terms with x = y in (6.14) vanish. On the other hand, each unordered pair x = y is encountered exactly twice.…”

Section: Now the Markov Property Implies Thatmentioning

confidence: 99%

Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

Ioffe

Velenik

Wachtel

2017

Probab. Theory Relat. Fields

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Abstract. We consider families of non-colliding random walks above a hard wall, which are subject to a self-potential of tilted area type. We view such ensembles as effective models for the level lines of a class of 2 + 1-dimensional discrete-height random surfaces in statistical mechanics. We prove that, under rather general assumptions on the step distribution and on the self-potential, such walks converge, under appropriate rescaling, to non-intersecting Ferrari-Spohn diffusions associated with limiting Sturm-Liouville operators. In particular, the limiting invariant measures are given by the squares of the corresponding Slater determinants.

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“…Under the stated hypotheses, the Mermin-Wagner theo rem and its successive generalizations [8][9][10][11][12][13][14][15][16][17][18] …”

Section: Available Mathematical Resultsmentioning

confidence: 99%

Comment on “Temperature-dependent orientational ordering on a spherical surface modeled with a lattice spin model”

Romano

2015

Phys. Rev. E

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In Sec. II of the named paper, the authors study the planar model on a regular two-dimensional (square or triangular) lattice and find evidence of an ordering transition at finite temperatures in both cases. It is shown that their findings do not agree with, and appear to ignore, a number of mathematical results, which have been known in the literature for some decades to date, and entail orientational disorder at all finite temperatures, as well as the existence of the Berezinski-Kosterlitz-Thouless transition.

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An Invariance Principle to Ferrari–Spohn Diffusions

Cited by 38 publications

References 17 publications

Airy distribution: Experiment, large deviations, and additional statistics

Airy distribution: Experiment, large deviations, and additional statistics

Dyson Ferrari–Spohn diffusions and ordered walks under area tilts

Comment on “Temperature-dependent orientational ordering on a spherical surface modeled with a lattice spin model”

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