Unitary matrix models and random partitions: Universality and multi-criticality

The generating functions for the gauge theory observables are often represented in terms of the unitary matrix integrals. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tracy-Widom distribution in the context of random partitions. We obtain the universal results for the multi-critical model in the weak and strong coupling phases. The free energy of the instanton sector in the weak coupling regime, and the genus expansion of the free energy in the strong coupling regime are explicitly computed and the universal multi-critical phase structure of the model is explored. Finally, we apply our results in concrete examples of supersymmetric indices of gauge theories in the large N limit.

Double Lusin Condition for the Ito-Henstock Integrable Operator-Valued Stochastic Process

In this paper, using double Lusin condition, we give an equivalent denition of the Ito-Henstock integral of an operator-valued stochastic process with respect to a Hilbert space-valued Wiener process.

ON BOUNDED WEAK AND PSEUDO-SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS HAVING TRICHOTOMY WITH AND WITHOUT DELAY IN BANACH SPACES

In the present work we consider E is a Banach space, E* is its dual space and L(E) is the space of continuous linear operators from E to itself. A function x: ℝ → E is said to be a pseudo-solution of the equation [Formula: see text] where A:ℝ → L(E) is strongly measurable and Bochner integrable function on every finite subinterval of ℝ with f:ℝ × E → E is only assumed to be weakly weakly sequentially continuous or Pettis-integrable and the linear equation [Formula: see text] has a trichotomy with constants α ≥ 1 and σ > 0, if x is absolutely continuous function and for each x* ∈ E* there exists a negligible set ℵx* such that for each t ∉ ℵx*, then we have [Formula: see text] We give an existence theorem for bounded weak and pseudo-solutions of the nonlinear differential equations [Formula: see text] Let T, r, d > 0, Br = {x > E: ‖x‖ ≤ r} and CE([-d,0]) be the Banach space of continuous functions from [-d,0] into E. Finally we prove an existence result for the differential equation with delay [Formula: see text] where fd : [a,b] × CE([-d,0]) → E is weakly weakly sequentially continuous function, [Formula: see text] is strongly measurable and Bochner integrable operator on [a,b] and θtx(s) = x(t + s) for all s ∈ [-d,0].

Discrete Tracy–Widom operators

Integrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.