Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910884947820544 |
|---|---|
| author | Mingo, James A. George, Daniel Munoz |
| author_facet | Mingo, James A. George, Daniel Munoz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17608 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices Mingo, James A. George, Daniel Munoz Probability 60B20 (Primary) 46L54, 15B52 (Secondary) 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. |
| title | Asymptotic limit of cumulants and higher order free cumulants of complex Wigner matrices |
| topic | Probability 60B20 (Primary) 46L54, 15B52 (Secondary) |
| url | https://arxiv.org/abs/2407.17608 |