In this paper we characterize the Nazarov-Sklyanin hierarchy for the classical periodic Benjamin-Ono equation in two complementary degenerations: for the multi-phase initial data (the periodic multi-solitons) at fixed dispersion and for bounded initial data in the limit of small dispersion. First, we express this hierarchy in terms of a piecewise-linear function of an auxiliary real variable which we call a dispersive action profile and whose regions of slope ±1 we call gaps and bands, respectively. Our expression uses Kerov's theory of profiles and Kreȋn's spectral shift functions. Next, for multi-phase initial data, we identify Baker-Akhiezer functions in Dobrokhotov-Krichever and Nazarov-Sklyanin and prove that multi-phase dispersive action profiles have finitely-many gaps determined by the singularities of their Dobrokhotov-Krichever spectral curves. Finally, for bounded initial data independent of the coefficient of dispersion, we show that in the small dispersion limit, the dispersive action profile concentrates weakly on a convex profile which encodes the conserved quantities of the dispersionless equation. To establish the weak limit, we reformulate Szegő's first theorem for Toeplitz operators using spectral shift functions. To illustrate our results, we identify the dispersive action profile of sinusoidal initial data with a profile found by Nekrasov-Pestun-Shatashvili and its small dispersion limit with the convex profile found by Vershik-Kerov and Logan-Shepp. CONTENTS

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In this paper we describe the spectrum of the quantum periodic Benjamin-Ono equation in terms of the multi-phase solutions of the underlying classical system (the periodic multi-solitons). To do so, we show that the semi-classical quantization of this system given by Abanov-Wiegmann is exact and equivalent to the geometric quantization by Nazarov-Sklyanin. First, for the Liouville integrable subsystems defined from the multi-phase solutions, we use a result of Gérard-Kappeler to prove that if one neglects the infinitely-many transverse directions in phase space, the regular Bohr-Sommerfeld conditions on the actions are equivalent to the condition that the singularities of the Dobrokhotov-Krichever multi-phase spectral curves define an anisotropic partition (Young diagram). Next, we show that the renormalization of the classical dispersion coefficient in Abanov-Wiegmann is implicit in the definition of the quantum Lax operator in Nazarov-Sklyanin. Finally, we verify that the regular Bohr-Sommerfeld conditions for the multi-phase solutions in the renormalized theory give the exact quantum spectrum determined by Nazarov-Sklyanin without any Maslov index correction.

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We derive exact and asymptotic results for random partitions from general results in the semi-classical analysis of coherent states. In geometric quantization of Hermitian affine spaces (M , J, g, ω), a coherent state Υ v (•| ) around a classical state v ∈ M is the reproducing kernel in the Fock space L 2 J-hol (M , dρ ,g ) of J-holomorphic functions on M square-integrable against the Segal-Bargmann Gaussian weight dρ ,g . Under certain regularity assumptions, in any canonical quantization defined by an ordering η, we prove that in the semi-classical limit → 01/2 with mean 0 and variance ||(∇O)| v || 2 g independent of η. These results do not assume integrability of the Hamiltonian flow generated by O but follow directly from the fact that at fixed > 0 the Segal-Bargmann weight on M is already Gaussian with covariance kernel g −1 given by the inverse metric.The classical periodic Benjamin-Ono equation for real 2π-periodic v of mean a ∈ R is Hamiltonian in the leaf (M (a), J, g −1/2 , ω −1/2 ) of the real L 2 -Sobolev space on the circle T at critical regularity s = −1/2 with J the spatial periodic Hilbert transform. We find a classical conserved density dF |v (c|ε) on c ∈ R for this system with dispersion coefficient ε, extending Nazarov-Sklyanin (2013). The authors also give an ordering η N S for an integrable canonical quantization, which we use to construct a quantum conserved density d F η N S (c| , ε)| Ψ . For quantum stationary states, we identify this conserved density with dF λ (c| ε 2 , ε 1 ) the Rayleigh measure of the profile of a partition λ of anisotropy (εAs Jack polynomials are the quantum stationary states and Stanley's Cauchy kernel (1989) is the reproducing kernel, the random values of the quantum periodic Benjamin-Ono hierarchy in a coherent state Υ v (•| ) are a "Jack measure" on partitions, a dispersive generalization of Okounkov's Schur measures (1999). By the above,we have concentration on a limit shape as → 0, the classical conserved density at v, and quantum fluctuations are an explicit Gaussian field. Our results follow from an enumerative asymptotic expansion in and ε of joint cumulants over new combinatorial objects we call "ribbon paths". As above, our results reflect the fact that at fixed > 0 the weight defining Fock space is already a fractional Brownian motion of variance and Hurst index (−s) − 1 2 dim T = + 1 2 − 1 2 = 0.