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Elliptic analogue of Vershik-Kerov limit shape

Andrey Grekov, Nikita Nekrasov

Abstract

We review the limit shape problem for the Plancherel measure and its generalizations found in supersymmetric gauge theory instanton count. We focus on the measure, interpolating between the Plancherel measure and uniform measure, a U(1) case of N=2* gauge theory. We give the formula for its limit shape in terms of elliptic functions, generalizing the trigonometric ``arcsin'' law of Vershik-Kerov and Logan-Schepp.

Elliptic analogue of Vershik-Kerov limit shape

Abstract

We review the limit shape problem for the Plancherel measure and its generalizations found in supersymmetric gauge theory instanton count. We focus on the measure, interpolating between the Plancherel measure and uniform measure, a U(1) case of N=2* gauge theory. We give the formula for its limit shape in terms of elliptic functions, generalizing the trigonometric ``arcsin'' law of Vershik-Kerov and Logan-Schepp.
Paper Structure (10 sections, 1 theorem, 112 equations, 6 figures)

This paper contains 10 sections, 1 theorem, 112 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Profiles, limit shapes, the arcsin law
  3. Elliptic generalization of Vershik-Kerov model
  4. Solving for limit shape in the elliptic case
  5. Limits and asymptotics
  6. Edge asymptotics
  7. Matching the edge behaviour.
  8. Expanding around Vershik-Kerov limir shape
  9. Conclusions and future directions
  10. Appendix. Proof of the factorization formula

Key Result

Theorem 2.1

(Vershik-Kerov-Logan-Schepp'77, VKLS) The limit shape of the distribution eq:Plancherel on the set of Young diagrams of size $N$ as $\hbar \sim \frac{\Lambda}{\sqrt{2N}} \rightarrow 0$ is described by the following profile: representing $\hbar$-rescaled piece-wise linear boundary of $\lambda$.

Figures (6)

  • Figure 1: Young diagram profile function
  • Figure 2: Re$[x(z)]$
  • Figure 3: $X(\theta)$
  • Figure 4: $X(\theta)$ with colored sides of the cut
  • Figure 5: Domain and image of $x(z)$
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 2.1