Concentration for integrable directed polymer models

Noack, Christian; Sosoe, Philippe

doi:10.48550/arxiv.2005.00126

Cited by 2 publications

(3 citation statements)

References 15 publications

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“…For example, prior to the present work, optimal-order bounds were available via coupling only for certain low central moments of these random variables. In a recent advance, refining the coupling method suitably, preprints [70,71] managed to establish nearly optimal (with an -deficiency in the exponents) bounds for all central moments of the free energies in the O'Connell-Yor polymer and the four basic integrable lattice polymers. As our work demonstrates, however, there is still significant room for further fundamental improvements to the method.…”

mentioning

confidence: 99%

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

Emrah¹,

Janjigian²,

Seppäläinen³

2021

Preprint

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We develop a new probabilistic method for deriving deviation estimates in directed planar polymer and percolation models. The key estimates are for exit points of geodesics as they cross transversal down-right boundaries. These bounds are of optimal cubic-exponential order. We derive them in the context of last-passage percolation with exponential weights with near-stationary boundary conditions. As a result, the probabilistic coupling method is empowered to treat a variety of problems optimally, which could previously be achieved only via inputs from integrable probability. As applications in the bulk setting, we obtain upper bounds of cubic-exponential order for transversal fluctuations of geodesics, and cube-root upper bounds with a logarithmic correction for distributional Busemann limits and competition interface limits. Several other applications are already in the literature.

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mentioning

confidence: 99%

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

Emrah¹,

Janjigian²,

Seppäläinen³

2021

Preprint

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“…This can be quantified by the occupation length j (defined above) which the polymer spends on column j. The statistics of this observable is one of the focus of this paper, and has not been addressed until very recently in the mathematics literature [57,58]. As we will see below the occupation fraction j /x plays the role of an order parameter for the localization transition.…”

Section: Definition Of the Model For A Single Linementioning

confidence: 99%

“…Note that the interpretation is that the occupation of the column a 1 is θ − θ c . We insert the Taylor expansion of ϕ 0 (z) around z = z * = a 1 inside formula (58), and change integration variable as…”

Section: Localized Phasementioning

confidence: 99%

Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization

Krajenbrink,

Doussal,

O'Connell

2020

Preprint

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We revisit the problem of an elastic line (such as a vortex line in a superconductor) subject to both columnar disorder and point disorder in dimension d = 1 + 1. Upon applying a transverse field, a delocalization transition is expected, beyond which the line is tilted macroscopically. We investigate this transition in the fixed tilt angle ensemble and within a "one-way" model where backward jumps are neglected. From recent results about directed polymers in the mathematics literature, and their connections to random matrix theory, we find that for a single line and a single strong defect this transition in presence of point disorder coincides with the Baik-Ben Arous-Péché (BBP) transition for the appearance of outliers in the spectrum of a perturbed random matrix in the Gaussian Unitary Ensemble. This transition is conveniently described in the polymer picture by a variational calculation. In the delocalized phase, the ground state energy exhibits Tracy-Widom fluctuations. In the localized phase we show, using the variational calculation, that the fluctuations of the occupation length along the columnar defect are described by fKPZ, a distribution which appears ubiquitously in the Kardar-Parisi-Zhang universality class. We then consider a smooth density of columnar defect energies. Depending on how this density vanishes at its lower edge we find either (i) a delocalized phase only (ii) a localized phase with a delocalization transition. We analyze this transition which is an infinite-rank extension of the BBP transition. The fluctuations of the ground state energy of a single elastic line in the localized phase (for fixed columnar defect energies) are described by a Fredholm determinant based on a new kernel, closely related to the kernel describing the largest real eigenvalues of the real Ginibre ensemble. The case of many columns and many non-intersecting lines, relevant for the study of the Bose glass phase, is also analyzed. The ground state energy is obtained using free probability and the Burgers equation. Connections with recent results on the generalized Rosenzweig-Porter model suggest that the localization of many polymers occurs gradually upon increasing their lengths.

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Concentration for integrable directed polymer models

Cited by 2 publications

References 15 publications

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization

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