Asymptotic expansion of spherical integral. (English) Zbl 1426.15057
The authors investigate the asymptotics of the spherical integral \[ I_{N}^{(\beta)}(\theta,B_N)=\int \exp\{\theta N(e_1^*B_Ne_1)\}dm_N^{(\beta)}(U),\;\beta=1,2, \] where \(m_N^{(\beta)}\) is the Haar measure on the orthogonal group O\((N)\) if \(\beta=1\) and on the unitary group U\((N)\) if \(\beta=2\), \(B_N\) is a deterministic \(N\times N\) real symmetric or Hermitian matrix and \(e_1\) is the first column of \(U\). The above spherical integral is in fact a finite-dimensional analog of the \(R\)-transform in free probability theory. The first and the second term in the asymptotic expansion are explicitly computed.
Reviewer: Nicolae Lupa (Timisoara)
MSC:
| 15B52 | Random matrices (algebraic aspects) |
| 15A16 | Matrix exponential and similar functions of matrices |
| 46L54 | Free probability and free operator algebras |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
| 43A90 | Harmonic analysis and spherical functions |
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