Asymptotics of Plancherel measures for symmetric groups

Borodin, Alexei; Okounkov, Andreĭ; Olshanski, Grigori

doi:10.1090/s0894-0347-00-00337-4

Cited by 346 publications

(386 citation statements)

References 29 publications

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“…We prove (3.14) in Appendix A. Amazingly, for the case ℓ = 0, such a summation formula appeared already in the problem of defining Z-measures on partitions [78][79][80][81][82][83][84] 5 . Note that this formula was also useful in the case of thermal correlation functions of the Bose gas [85].…”

Section: Summation Of Critical Form Factorsmentioning

confidence: 98%

A form factor approach to the asymptotic behavior of correlation functions in critical models

Kitanine¹,

Kozłowski²,

Maillet³

et al. 2011

J. Stat. Mech.

125

184

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We propose a form factor approach for the computation of the large distance asymptotic behavior of correlation functions in quantum critical (integrable) models. In the large distance regime we reduce the summation over all excited states to one over the particle/hole excitations lying on the Fermi surface in the thermodynamic limit. We compute these sums, over the so-called critical form factors, exactly. Thus we obtain the leading large distance behavior of each oscillating harmonic of the correlation function asymptotic expansion, including the corresponding amplitudes. Our method is applicable to a wide variety of integrable models and yields precisely the results stemming from the Luttinger-liquid approach, the conformal field theory predictions and our previous analysis of the correlation functions from their multiple-integral representations. We argue that our scheme applies to a general class of non-integrable quantum critical models as well.

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Section: Summation Of Critical Form Factorsmentioning

confidence: 98%

A form factor approach to the asymptotic behavior of correlation functions in critical models

Kitanine¹,

Kozłowski²,

Maillet³

et al. 2011

J. Stat. Mech.

125

184

View full text Add to dashboard Cite

show abstract

“…Both of these sums can be computed explicitly via the identity used in [1] (see also [4][5][6][7][8][9][10]):…”

Section: General Schemementioning

confidence: 99%

“…It turns out that the corresponding series over these critical states and form factors can be summed up thanks to a beautiful (purely combinatorial) multiple sum formula (see [1]). It is worth mentioning that this summation formula has a deep group theoretical origin in the representation theory of the infinite permutation group [4][5][6][7][8][9][10]. In the context of critical models, it gives the explicit correspondence between the microscopic dynamics and its critical (conformal) limit.…”

Section: Introductionmentioning

confidence: 99%

Form factor approach to dynamical correlation functions in critical models

Kitanine¹,

Kozłowski²,

Maillet³

et al. 2012

J. Stat. Mech.

115

167

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We develop a form factor approach to the study of dynamical correlation functions of quantum integrable models in the critical regime. As an example, we consider the quantum non-linear Schrödinger model. We derive long-distance/long-time asymptotic behavior of various two-point functions of this model. We also compute edge exponents and amplitudes characterizing the power-law behavior of dynamical response functions on the particle/hole excitation thresholds. These last results confirm predictions based on the non-linear Luttinger liquid method. Our results rely on a first principles derivation, based on the microscopic analysis of the model, without invoking, at any stage, some correspondence with a continuous field theory. Furthermore, our approach only makes use of certain general properties of the model, so that it should be applicable, with possibly minor modifications, to a wide class of (not necessarily integrable) gapless one dimensional Hamiltonians.

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“…This is typically the case of discrete measures that appear in sums over Young diagrams [11,12,13,14,15,16,17,18].…”

Section: Properties Of the Determinant Formulamentioning

confidence: 99%

Six-vertex model with domain wall boundary conditions and one-matrix model

2000

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The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a hermitean one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the six-vertex model with DWBC in the different phases. In particular, in the anti-ferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions.

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Asymptotics of Plancherel measures for symmetric groups

Cited by 346 publications

References 29 publications

A form factor approach to the asymptotic behavior of correlation functions in critical models

A form factor approach to the asymptotic behavior of correlation functions in critical models

Form factor approach to dynamical correlation functions in critical models

Six-vertex model with domain wall boundary conditions and one-matrix model

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