The Continuum Directed Random Polymer

Alberts, Tom; Khanin, Konstantin; Quastel, Jeremy

doi:10.1007/s10955-013-0872-z

Cited by 93 publications

(179 citation statements)

References 20 publications

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“…the three motivating models listed above). We stress here that the difficulty is substantial and not just technical: for pinning and directed polymer models, one can show [AKQ14b,CSZ14] that the scaling limit P W Ω;λ,ĥ of P ω Ω δ ;λ,h exists, but forλ = 0 it is not absolutely continuous with respect to P ref Ω . In particular, it is hopeless to define the continuum disordered model through a Radon-Nikodym density, like in (1.4).…”

Section: Continuum Limits Of Disordered Systemsmentioning

confidence: 96%

“…Similar to the case α = 2 studied in [AKQ14b], we expect that this convergence can be upgraded to a convergence in distribution in the space of continuous functions, equipped with the uniform topology. We can then use these partition functions to define a continuum long-range directed polymer model (which corresponds intuitively to an "α-stable Lévy process in a white noise random medium"), by specifying its finite-dimensional distributions as done in [AKQ14b] for the Brownian case α = 2.…”

Section: Scaling Limits Of Disordered Systemsmentioning

confidence: 99%

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Polynomial chaos and scaling limits of disordered systems

Caravenna¹,

Sun²,

Zygouras³

2016

J. Eur. Math. Soc.

140

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Abstract. Inspired by recent work of Alberts, Khanin and Quastel [AKQ14a], we formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables (called polynomial chaos expansions) converges to a limiting random variable, given by a Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz [MOO10]. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the (long-range) directed polymer model in dimension 1 + 1, and the two-dimensional random field Ising model. This gives a new perspective in the study of disorder relevance, and leads to interesting new continuum models that warrant further studies.

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Section: Continuum Limits Of Disordered Systemsmentioning

confidence: 96%

Section: Scaling Limits Of Disordered Systemsmentioning

confidence: 99%

Polynomial chaos and scaling limits of disordered systems

Caravenna¹,

Sun²,

Zygouras³

2016

J. Eur. Math. Soc.

140

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“…We will restrict ourselves here to the one space dimension with the quadratic non-linearity (1). Even in this 1 + 1 dimensional situation we are still in the very difficult case of a field theory with broken time reversible invariance.…”

Section: ) ξ(T X) ξ(S Y) = δ(T − S)δ(x − Y)mentioning

confidence: 99%

“…Traditionally it has a square root so that D is the mean square, or variance. The key term (1), the deterministic part of the growth, is assumed to be a function only of the slope, and to be a symmetric function. Here is a picture of what we mean by lateral growth From the picture, the natural choice for F might be (1+|∂ x h| 2 ) −1/2 , however this leads to a seemingly intractable equation.…”

Section: (T X) ξ(S Y) := E[ξ(t X)ξ(s Y)] = δ(T − S)δ(x − Y) √mentioning

confidence: 99%

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Introduction to KPZ

Quastel

2011

Current Developments in Mathematics

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177

140

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Abstract. This is an introductory survey of the Kardar-Parisi-Zhang equation (KPZ). The first chapter provides a non-rigorous background to the equation and to some of the many models which are supposed to lie in its universality class, as well as the predicted, non-standard fluctuations. The second chapter provides a rigorous introduction to the stochastic heat equation, whose logarithm is the solution of KPZ, as well as some of the known methods for proving convergence of discrete growth models and directed polymer free energies. Finally, we end with a sketch of the derivation of exact formulas for the one-point distributions of KPZ at finite time for special initial data, from the Tracy-Widom formulas for asymmetric exclusion.

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Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension

Borodin

Corwin

Ferrari

2014

Comm Pure Appl Math

158

263

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We consider two models for directed polymers in space-time independent random media (the O'Connell-Yor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time , the probability distributions for the free energy fluctuations, when rescaled by 1=3 , converges to the GUE Tracy-Widom distribution.We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semidiscrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik-Ben Arous-Péché distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm-the Kardar-Parisi-Zhang equation. The Laplace transform formula we prove can be inverted to give the one-point probability distribution of the solution to these stochastic PDEs for the class of initial data.

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The Continuum Directed Random Polymer

Cited by 93 publications

References 20 publications

Polynomial chaos and scaling limits of disordered systems

Polynomial chaos and scaling limits of disordered systems

Introduction to KPZ

Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension

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