Exponential of the \(S^1\) trace of the free field and Verblunsky coefficients. (English) Zbl 1493.60015
Summary: An identity of Szego, and a volume calculation, heuristically suggest a simple expression for the distribution of Verblunsky coefficients with respect to the (normalized) exponential of the \(S^1\) trace of the Gaussian free field. This heuristic expression is not quite correct. A proof of the correct formula has been found by R. Chhaibi and J. Najnudel [“On the cirle, \(GMC^\gamma=\underleftarrow{\lim}\,C\beta E_n\) for \(\gamma=\sqrt{\frac{2}{\beta}}\), \((\gamma\leq 1)\)”, Preprint, arXiv:1904.00578]. Their proof uses random matrix theory and overcomes many difficult technical issues. In addition to presenting the Szego perspective, we show that the Chhaibi and Najnudel theorem implies a family of combinatorial identities (for moments of measures) which are of intrinsic interest.
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
| 60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |
| 60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |
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