### Abstract

We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik [17], are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the so-called Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams.

Original language | English |
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Pages (from-to) | 335-363 |

Number of pages | 29 |

Journal | Communications in Mathematical Physics |

Volume | 299 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*299*(2), 335-363. https://doi.org/10.1007/s00220-010-1082-z

**Hydrodynamic Limit for an Evolutional Model of Two-Dimensional Young Diagrams.** / Funaki, Tadahisa; Sasada, Makiko.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 299, no. 2, pp. 335-363. https://doi.org/10.1007/s00220-010-1082-z

}

TY - JOUR

T1 - Hydrodynamic Limit for an Evolutional Model of Two-Dimensional Young Diagrams

AU - Funaki, Tadahisa

AU - Sasada, Makiko

PY - 2010

Y1 - 2010

N2 - We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik [17], are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the so-called Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams.

AB - We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik [17], are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the so-called Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams.

UR - http://www.scopus.com/inward/record.url?scp=77955846353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955846353&partnerID=8YFLogxK

U2 - 10.1007/s00220-010-1082-z

DO - 10.1007/s00220-010-1082-z

M3 - Article

AN - SCOPUS:77955846353

VL - 299

SP - 335

EP - 363

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -