Random polymers via orthogonal Whittaker and symplectic Schur functions

Bisi, Elia

doi:10.48550/arxiv.1810.03734

Cited by 5 publications

(6 citation statements)

References 3 publications

(5 reference statements)

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“…This process can be represented by Figure 1 where the interaction → is now reflection and the interaction is now a weighted indicator function. The zero-temperature limit of the field of log partition functions is the field of point-to-line last passage percolation times {G(i, j) : i + j ≤ n + 1} (see [6,7]) and it is natural to expect that {G(i, j) : i + j ≤ n + 1} is the invariant measure of {X ij : i+j ≤ n+1}. However, we do not prove this because the discontinuities in the drifts means that the conditions for Lemma 14 are no longer satisfied.…”

Section: Zero-temperature Limitsmentioning

confidence: 99%

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Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

FitzGerald

Warren

2020

Probab. Theory Relat. Fields

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This paper proves an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-toline partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne's identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field of point-to-line log partition functions for the log-gamma polymer.

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Section: Zero-temperature Limitsmentioning

confidence: 99%

“…where {W (β) ij : i + j ≤ n + 1} are random variables with inverse gamma distributions and rates β −1 (α i + β j ). As β → ∞, the left hand side converges almost surely by Laplace's Theorem and the right hand side converges by [6,7] to give, sup 0=s0≤...≤sn<∞…”

Section: Zero-temperature Limitsmentioning

confidence: 99%

Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

FitzGerald

Warren

2020

Probab. Theory Relat. Fields

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“…Furthermore, such a piecewise linear description is naturally prone to be extended to generic Young-diagram-shaped input arrays (not necessarily rectangular). The latter aspect was useful to study last passage percolation models with point-to-line and point-to-half-line path geometries and/or various symmetries on the input weights: this lead to new exact formulas for such models in terms of all the irreducible characters of the classical groups, in particular symplectic and orthogonal Schur polynomials [Bis18;BZ19b;BZ19c]. The Burge correspondence is, in the classical combinatorial description, a variant of the RSK mapping that bijectively transforms a non-negative integer matrix into a pair of semistandard Young tableaux of the same shape, through a column insertion algorithm.…”

Section: Introductionmentioning

confidence: 99%

The geometric Burge correspondence and the partition function of polymer replicas

Bisi,

O'Connell,

Zygouras

2020

Preprint

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We construct a geometric lifting of the Burge correspondence as a composition of local birational maps on generic Young-diagram-shaped arrays. We prove a fundamental link with the geometric Robinson-Schensted-Knuth correspondence and with the geometric Sch ützenberger involution. We also show a number of properties of the geometric Burge correspondence, also specializing them to the case of symmetric input arrays. In particular, our construction shows that such a mapping is volume preserving in log-log variables. As an application, we consider a model of two polymer paths of given length constrained to have the same endpoint, known as polymer replica. We prove that the distribution of the polymer replica partition function in a log-gamma random environment is a Whittaker measure, and deduce the corresponding Whittaker integral identity. For a certain choice of the parameters, we notice a distributional identity between our model and the symmetric log-gamma polymer studied by O'Connell, Seppäläinen, and Zygouras

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“…(2.5) † Notice that (1.11) and (1.12) have now been obtained directly via the standard Robinson-Schensted-Knuth correspondence, avoiding the route of the zero temperature limit, see [Bis18,BZ19b].…”

Section: From Determinants To Fredholm Determinantsmentioning

confidence: 99%

GOE and $${\mathrm{Airy}}_{2\rightarrow 1}$$ Marginal Distribution via Symplectic Schur Functions

Bisi

Zygouras

2019

Springer Proceedings in Mathematics &Amp; Statistics

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We derive Sasamoto's Fredholm determinant formula for the Tracy-Widom GOE distribution, as well as the one-point marginal distribution of the Airy 2→1 process, originally derived by Borodin-Ferrari-Sasamoto, as scaling limits of point-to-line and point-to-half-line directed last passage percolation with exponentially distributed waiting times. The asymptotic analysis goes through new expressions for the last passage times in terms of integrals of (the continuous analog of) symplectic and classical Schur functions, obtained recently in [BZ19a].2010 Mathematics Subject Classification. Primary: 60Cxx, 05E05, 82B23; Secondary: 11Fxx, 82D60. Key words and phrases. Point-to-line directed last passage percolation, Tracy-Widom GOE distribution, Airy 2→1 , Schur functions, symplectic Schur functions. † Notice that (1.11) and (1.12) have now been obtained directly via the standard Robinson-Schensted-Knuth correspondence, avoiding the route of the zero temperature limit, see [Bis18,BZ19b].

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Random polymers via orthogonal Whittaker and symplectic Schur functions

Cited by 5 publications

References 3 publications

Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

Point-to-line last passage percolation and the invariant measure of a system of reflecting Brownian motions

The geometric Burge correspondence and the partition function of polymer replicas

GOE and $${\mathrm{Airy}}_{2\rightarrow 1}$$ Marginal Distribution via Symplectic Schur Functions

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