Tracy-Widom Asymptotics for a River Delta Model

Barraquand, Guillaume; Rychnovsky, Mark

doi:10.1007/978-3-030-15096-9_17

Cited by 5 publications

(2 citation statements)

References 62 publications

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“…In the first inequality we used (2.11). In the second inequality we multiplied each term of the sum by m. In the third inequality, we use [8,Lemma 4.4] with C = (t 1/3 LR 3 ).…”

Section: )mentioning

confidence: 99%

Large deviations for sticky Brownian motions

Barraquand¹,

Rychnovsky²

2020

Electron. J. Probab.

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We consider n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability.These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a (previously known) limit of the exactly solvable beta random walk in random environment.

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“…In the first inequality we used (2.11). In the second inequality we multiplied each term of the sum by m. In the third inequality, we use [8,Lemma 4.4] with C = (t 1/3 LR 3 ).…”

Section: )mentioning

confidence: 99%

Large deviations for sticky Brownian motions

Barraquand¹,

Rychnovsky²

2020

Electron. J. Probab.

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“…polymer of [24]; • (α, β ) = (1, −1) corresponds to Bernoulli-exponential/geometric FPP, which was introduced as a zero-temperature limit of the β -RWRE in [4], and has also been called the river delta model (see [5]).…”

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confidence: 99%

On the stationary solutions of random polymer models and their zero-temperature limits

Croydon,

Sasada

2021

Preprint

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We derive stationary measures for certain zero-temperature random polymer models, which we believe are new in the case of the zero-temperature limit of the beta random polymer (that has been called the river delta model). To do this, we apply techniques developed for understanding the stationary measures of the corresponding positive-temperature random polymer models (and some deterministic integrable systems). More specifically, the article starts with a survey of results for the four basic beta-gamma models (i.e. the inverse-gamma, gamma, inverse-beta and beta random polymers), highlighting how the maps underlying the systems in question can each be reduced to one of two basic bijections, and that through an 'independence preservation' property, these bijections characterise the associated stationary measures. We then derive similar descriptions for the corresponding zero-temperature maps, whereby each is written in terms of one of two bijections. One issue with this picture is that, unlike in the positivetemperature case, the change of variables required is degenerate in general, and so whilst the argument yields stationary solutions, it does not provide a complete characterisation of them. We also derive from our viewpoint various links between random polymer models, some of which recover known results, some of which are novel, and some of which lead to further questions.

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Correction to: Random-walk in Beta-distributed random environment

Barraquand

Corwin

2022

Probab. Theory Relat. Fields

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Tracy-Widom Asymptotics for a River Delta Model

Cited by 5 publications

References 62 publications

Large deviations for sticky Brownian motions

Large deviations for sticky Brownian motions

On the stationary solutions of random polymer models and their zero-temperature limits

Correction to: Random-walk in Beta-distributed random environment

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