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Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices

James A. Mingo, Daniel Munoz George

Abstract

We compute the fluctuation moments $α_{m_1,\dots,m_r}$ of a Complex Wigner Matrix $X_N$ given by the limit $\lim_{N\rightarrow\infty}N^{r-2}k_r(Tr(X_N^{m_1}),\dots,Tr(X_N^{m_r}))$. We prove the limit exists and characterize the leading order via planar graphs that result to be trees. We prove these graphs can be counted by the set of non-crossing partitioned permutations which permit us to express the moments $α_{m_1,\dots,m_r}$ in terms of simpler quantities $κ_{m_1,\dots,m_r}$ known as the higher order cumulants. As for lower order dimensions ($r \leq 3$) we observe that while the moments have a more elaborated expression the cumulants are simpler.

Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices

Abstract

We compute the fluctuation moments of a Complex Wigner Matrix given by the limit . We prove the limit exists and characterize the leading order via planar graphs that result to be trees. We prove these graphs can be counted by the set of non-crossing partitioned permutations which permit us to express the moments in terms of simpler quantities known as the higher order cumulants. As for lower order dimensions () we observe that while the moments have a more elaborated expression the cumulants are simpler.
Paper Structure (13 sections, 29 theorems, 214 equations, 8 figures, 1 table)

This paper contains 13 sections, 29 theorems, 214 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Non-crossing sets
  3. Non-crossing partitions and free cumulants
  4. Non-crossing permutations
  5. Non-crossing partitioned permutations and higher order free cumulants
  6. Graph theory
  7. Quotient graphs by identification of the edges
  8. Asymptotic expansion of the cumulants
  9. Characterization of the leading order
  10. Non-crossing limit triples
  11. Crossing limit triples
  12. Higher order cumulants
  13. Computation of lower order free cumulants

Key Result

Theorem 1.5

Let $X_N$ be a complex Wigner matrix with moment sequence give by the limit of Equation $(Equation: definition of higher order moments)$. Let $u,v\geq 0$ be non-negative integers such that $u$ is even and let $r=u/2+v$. Let $(\delta_r)_r$ be the sequence given by the recursive equation $\delta_1=\be for $r\geq 2$. Then the higher order free cumulants of $X_N$ are give by where there are exactly $

Figures (8)

  • Figure 1: The oriented graphs of Example \ref{['Example: Oriented partition of the edges']}.
  • Figure 2: The graph $G(\tau)$ corresponding to $G=(V,E)$ and $\tau$ given as in Example \ref{['Example: Oriented partition of the edges']}.
  • Figure 3: The graph $T$ corresponding to $m_1=4$ and $m_2=m_3=m_4=m_5=m_6=m_7=2$.
  • Figure 4: The graphs of Example \ref{['Example: Example of limit and non-limit partition']}
  • Figure 5: The graphs of Example \ref{['Example: Big example']}
  • ...and 3 more figures

Theorems & Definitions (87)

  • Definition 1.1
  • Definition 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.4
  • Proposition 2.6: Triangle Inequality
  • Proposition 2.7: Equality in the triangle inequality
  • Remark 2.8
  • ...and 77 more