Latifi, Mohammad Javad; Pickrell, Doug Exponential of the \(S^1\) trace of the free field and Verblunsky coefficients. (English) Zbl 1493.60015 Rocky Mt. J. Math. 52, No. 3, 899-924 (2022). 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. Cited in 1 Document 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 Keywords:Verblunsky coefficients; Gaussian free field; loop group factorization × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link References: [1] K. Astala, P. Jones, A. Kupiainen, and E. Saksman, “Random conformal weldings”, Acta Math. 207:2 (2011), 203-254. · Zbl 1253.30032 · doi:10.1007/s11511-012-0069-3 [2] E. Basor and D. Pickrell, “Loops in \[{\rm SL}(2,\mathbb C)\] and root subgroup factorization”, Random Matrices Theory Appl. 7:3 (2018), 1850008, 26. · Zbl 1402.22022 · doi:10.1142/S2010326318500089 [3] N. Berestycki, “An elementary approach to Gaussian multiplicative chaos”, Electron. Commun. Probab. 22 (2017), Paper No. 27, 12. · Zbl 1365.60035 · doi:10.1214/17-ECP58 [4] A. Caine and D. Pickrell, “Loops in noncompact groups of Hermitian symmetric type and factorization”, J. Gen. Lie Theory Appl. 9:2 (2015), Art. ID 1000233, 14. · Zbl 1395.53063 · doi:10.4172/1736-4337.1000233 [5] R. Chhaibi and J. Najnudel, “On the circle, \[GMC^{\gamma} = \varprojlim C\beta E_n\] for \[\gamma= \sqrt{2/\beta} , \gamma\leq 1 \]”, preprint, 2019. · Zbl 1406.60075 [6] J.-P. Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics 5, Cambridge University Press, 1985. · Zbl 0571.60002 [7] S. Kerov, G. Olshanski, and A. Vershik, “Harmonic analysis on the infinite symmetric group”, Invent. Math. 158:3 (2004), 551-642. · Zbl 1057.43005 · doi:10.1007/s00222-004-0381-4 [8] S. Lang, \[{\rm SL}_2(\mathbb{R})\], Addison-Wesley, Reading, MA, 1975. · Zbl 0311.22001 [9] I. E. Segal, “Ergodic subgroups of the orthogonal group on a real Hilbert space”, Ann. of Math. (2) 66 (1957), 297-303. · Zbl 0083.10603 · doi:10.2307/1970001 [10] B. Simon, “OPUC on one foot”, Bull. Amer. Math. Soc. (N.S.) 42:4 (2005), 431-460. · Zbl 1108.42005 · doi:10.1090/S0273-0979-05-01075-X [11] B. Simon, Orthogonal polynomials on the unit circle: part 1, American Mathematical Society Colloquium Publications 54, American Mathematical Society, Providence, RI, 2005. · Zbl 1082.42021 · doi:10.1090/coll054.1 [12] M. Sodin, “Zeroes of Gaussian analytic functions”, pp. 445-458 in European Congress of Mathematics (Stockholm, June 27-July 2, 2004), European Mathematical Society Publishing House. · Zbl 1073.60058 · doi:10.4171/009-1/27 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.