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Free field theory and observables of periodic Macdonald processes
Combinatorics · Submitted 14 Jan 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory.
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The abstract, sentence by sentence
We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory. We also study several families of observables for periodic Macdonald processes and give formulas of their moments. The technical tool is the free field realization of operators that are diagonalized by the Macdonald symmetric functions, where the operators admit expressions by means of vertex operators. We show that, when we adopt Plancherel specializations, the corresponding periodic Macdonald process is related to a Young diagram-valued periodic continuous process. It is known that, for periodic Schur processes, determinantal formulas are only available when we extend them to take into account the charge. We also consider this kind of shift-mixed periodic Macdonald processes and discuss their Schur-limit.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. | contribution 0.99 | 0.86 | 0.14 | asserted 0.92 |
| s1 | It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. | background 1.00 | 0.04 | 0.03 | asserted 1.00 |
| s2 | We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory. | contribution 0.54 | 0.96 | 0.31 | asserted 1.00 |
| s3 | We also study several families of observables for periodic Macdonald processes and give formulas of their moments. | contribution 0.84 | 0.96 | 0.19 | asserted 1.00 |
| s4 | The technical tool is the free field realization of operators that are diagonalized by the Macdonald symmetric functions, where the operators admit expressions by means of vertex operators. | method 1.00 | 0.42 | 0.06 | asserted 0.99 |
| s5 | We show that, when we adopt Plancherel specializations, the corresponding periodic Macdonald process is related to a Young diagram-valued periodic continuous process. | result 0.95 | 0.93 | 0.12 | asserted 1.00 |
| s6 | It is known that, for periodic Schur processes, determinantal formulas are only available when we extend them to take into account the charge. | background 0.94 | 0.04 | 0.03 | asserted 1.00 |
| s7 | We also consider this kind of shift-mixed periodic Macdonald processes and discuss their Schur-limit. | contribution 0.99 | 0.88 | 0.06 | asserted 0.88 |
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The paper
Original abstract
We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic Schur processes and a periodic analogue of Macdonald processes. It is known that, in the theory of stochastic processes related to a family of symmetric functions, the Cauchy-like identity gives an explicit expression of a partition function. We compute the partition functions of periodic Macdonald processes relying on the free field realization of the Macdonald theory. We also study several families of observables for periodic Macdonald processes and give formulas of their moments. The technical tool is the free field realization of operators that are diagonalized by the Macdonald symmetric functions, where the operators admit expressions by means of vertex operators. We show that, when we adopt Plancherel specializations, the corresponding periodic Macdonald process is related to a Young diagram-valued periodic continuous process. It is known that, for periodic Schur processes, determinantal formulas are only available when we extend them to take into account the charge. We also consider this kind of shift-mixed periodic Macdonald processes and discuss their Schur-limit.
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References
How the literature uses it
1 arXiv paper cite this in 2 sentences
Papers citing it
Uses its method or tool 1 paper
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The formulas for the joint moments were inspired by [ref] , which obtained formulas for observables for periodic Macdonald processes.
Lozenge tilings and the Gaussian free field on a cylinder · Methods · 0.83
Compares with it 1 paper
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While the results of [ref] do not imply our formulas, there is agreement between the two in certain special cases.
Lozenge tilings and the Gaussian free field on a cylinder · Methods · 1.00
What it cites · 5 sentences without a confident label
Uses the method or tool of 1 paper
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In this paper, we apply the method in [ref] to periodic Macdonald processes.
Free field approach to the Macdonald process · Introduction · 0.80
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