Pfaffian Stochastic Dynamics of Strict Partitions

We study a family of continuous time Markov jump processes on strict partitions (partitions with distinct parts) preserving the distributions introduced by Borodin (1997) in connection with projective representations of the infinite symmetric group. The one-dimensional distributions of the processes (i.e., the Borodin's measures) have determinantal structure. We express the dynamical correlation functions of the processes in terms of certain Pfaffians and give explicit formulas for both the static and dynamical correlation kernels using the Gauss hypergeometric function. Moreover, we are able to express our correlation kernels (both static and dynamical) through those of the z-measures on partitions obtained previously by Borodin and Olshanski in a series of papers. The results about the fixed time case were announced in the author's note arXiv:1002.2714. A part of the present paper contains proofs of those results.Comment: AMS-LaTeX, 59 pages. v2: Added new results about connections with the z-measures and orthogonal spectral projection operators. Investigated asymptotic behaviour of the dynamical Pfaffian correlation kernel. Removed double contour integral expressions for correlation kernels to shorten the tex

The class of n-entire operators

We introduce a classification of simple, regular, closed symmetric operators with deficiency indices (1,1) according to a geometric criterion that extends the classical notions of entire operators and entire operators in the generalized sense due to M. G. Krein. We show that these classes of operators have several distinctive properties, some of them related to the spectra of their canonical selfadjoint extensions. In particular, we provide necessary and sufficient conditions on the spectra of two canonical selfadjoint extensions of an operator for it to belong to one of our classes. Our discussion is based on some recent results in the theory of de Branges spaces.Comment: 33 pages. Typos corrected. Changes in the wording of Section 2. References added. Examples added. arXiv admin note: text overlap with arXiv:1104.476

Quantum Spectral Curve for the eta-deformed AdS_5xS^5 superstring

The spectral problem for the ${\rm AdS}_5\times {\rm S}^5$ superstring and its dual planar maximally supersymmetric Yang-Mills theory can be efficiently solved through a set of functional equations known as the quantum spectral curve. We discuss how the same concepts apply to the $\eta$-deformed ${\rm AdS}_5\times {\rm S}^5$ superstring, an integrable deformation of the ${\rm AdS}_5\times {\rm S}^5$ superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the ${\rm AdS}_5\times {\rm S}^5$ superstring, like the relation between the XXZ and XXX spin chains, or the sausage and the ${\rm S}^2$ sigma models for instance. We derive the quantum spectral curve for the $\eta$-deformed string by reformulating the corresponding ground-state thermodynamic Bethe ansatz equations as an analytic $Y$ system, and map this to an analytic $T$ system which upon suitable gauge fixing leads to a $\mathbf{P} \mu$ system -- the quantum spectral curve. We then discuss constraints on the asymptotics of this system to single out particular excited states. At the spectral level the $\eta$-deformed string and its quantum spectral curve interpolate between the ${\rm AdS}_5\times {\rm S}^5$ superstring and a superstring on "mirror" ${\rm AdS}_5\times {\rm S}^5$, reflecting a more general relationship between the spectral and thermodynamic data of the $\eta$-deformed string. In particular, the spectral problem of the mirror ${\rm AdS}_5\times {\rm S}^5$ string, and the thermodynamics of the undeformed ${\rm AdS}_5\times {\rm S}^5$ string, are described by a second rational limit of our trigonometric quantum spectral curve, distinct from the regular undeformed limit.Comment: 32+37 pages; 6 figures. v2: added reference

Generalized Hermite processes, discrete chaos and limit theorems

We introduce a broad class of self-similar processes $\{Z(t),t\ge 0\}$ called generalized Hermite process. They have stationary increments, are defined on a Wiener chaos with Hurst index $H\in (1/2,1)$, and include Hermite processes as a special case. They are defined through a homogeneous kernel $g$, called "generalized Hermite kernel", which replaces the product of power functions in the definition of Hermite processes. The generalized Hermite kernels $g$ can also be used to generate long-range dependent stationary sequences forming a discrete chaos process $\{X(n)\}$. In addition, we consider a fractionally-filtered version $Z^\beta(t)$ of $Z(t)$, which allows $H\in (0,1/2)$. Corresponding non-central limit theorems are established. We also give a multivariate limit theorem which mixes central and non-central limit theorems.Comment: Corrected some error