Geometric RSK and the Toda lattice

O’Connell, Neil

doi:10.1215/ijm/1415023516

Cited by 12 publications

(11 citation statements)

References 47 publications

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“…Thus, if Y (0) = Y * (λ, X ) and we let Y (t) evolve according to(8.9), then Y N (t), t ≥ 0 is a realisation of the N -particle non-Abelian Toda flow on P N , with Y N (0) = X and C (N ) k = i λ k i I . In the scalar case, this agrees with the statement of[35, Theorem 8.4].…”

supporting

confidence: 89%

“…There is also an indefinite version, in which the potential has the opposite sign, as considered by Popowicz [40,41]. As in the scalar case [35,37], it is the indefinite version which is relevant to our setting. Writing A i = X i+1 X −1 i and B i =Ẋ i X −1 i , the Hamiltonian is given by…”

Section: The Non-abelian Toda Latticementioning

confidence: 98%

“…10) with the conventions, as above, Y m−0. It corresponds precisely to the drift term of the diffusion in T with infinitesimal generator G λ /2.It follows from (8.6) that T λ is invariant under the evolution (8.9), as in the scalar (n = 1) case[35, Proposition 8.1]. Thus, if Y (0) = Y * (λ, X ) and we let Y (t) evolve according to(8.9), then Y N (t), t ≥ 0 is a realisation of the N -particle non-Abelian Toda flow on P N , with Y N (0) = X and C (N ) k = i λ k i I .…”

mentioning

confidence: 97%

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Interacting diffusions on positive definite matrices

O’Connell

2021

Probab. Theory Relat. Fields

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We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to K-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda lattice.

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supporting

confidence: 89%

Section: The Non-abelian Toda Latticementioning

confidence: 98%

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confidence: 97%

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Interacting diffusions on positive definite matrices

O’Connell

2021

Probab. Theory Relat. Fields

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“…The geometric RSK correspondences are also sometimes called tropical RSK correspondences [44], [53], [22], despite the fact that they come from the process of detropicalization. We adopt a convention of calling them the geometric RSK correspondences (following, e.g., [60], [19], [57]). The latter name arises in connection with geometric crystals (see [19] for more background).…”

Section: Polymer Limits Of (α) Dynamics On Q-whittaker Processesmentioning

confidence: 99%

“…The geometric version (also sometimes called "tropical ") of the RS and the RSK correspondences 1 has also been employed in the study of stochastic systems [56], [22], [60], [57]. The systems one obtains at this level are related to directed random polymers in random media, in particular, to the O'Connell-Yor, log-Gamma, and strict-weak random polymers introduced in [61], [70], and [58], [25], respectively.…”

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confidence: 99%

$q$-randomized Robinson–Schensted–Knuth correspondences and random polymers

Матвеев¹,

Petrov²

2016

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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We introduce and study q-randomized Robinson-Schensted-Knuth (RSK) correspondences which interpolate between the classical (q = 0) and geometric (q 1) RSK correspondences (the latter ones are sometimes also called tropical).For 0 < q < 1 our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a certain class of probability measures on arrays, namely, on q-Whittaker processes (which are t = 0 versions of Macdonald processes of Borodin-Corwin [7]). We present four Markov dynamics which for q = 0 reduce to the classical row or column RSK correspondences applied to a random input matrix with independent geometric or Bernoulli entries.Our new two-dimensional discrete time dynamics generalize and extend several known constructions: (1) The discrete time q-TASEPs studied by Borodin-Corwin [8] arise as one-dimensional marginals of our "column" dynamics. In a similar way, our "row" dynamics lead to discrete time q-PushTASEPs -new integrable particle systems in the Kardar-Parisi-Zhang universality class. We employ these new one-dimensional discrete time systems to establish a Fredholm determinantal formula for the two-sided continuous time q-PushASEP conjectured by Corwin-Petrov [23]. (2) In a certain Poisson-type limit (from discrete to continuous time), our two-dimensional dynamics reduce to the q-randomized column and row Robinson-Schensted correspondences introduced by O'Connell-Pei [59] and Borodin-Petrov [16], respectively. (3) In a scaling limit as q 1, two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma (introduced by Seppäläinen [70]) or strict-weak (introduced independently by O'Connell-Ortmann [58] and Corwin-Seppäläinen-Shen [25]) directed random lattice polymers.

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Stochastic Bäcklund Transformations

O’Connell

2015

Lecture Notes in Mathematics

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Geometric RSK and the Toda lattice

Cited by 12 publications

References 47 publications

Interacting diffusions on positive definite matrices

Interacting diffusions on positive definite matrices

$q$-randomized Robinson–Schensted–Knuth correspondences and random polymers

Stochastic Bäcklund Transformations

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