On explicit free field realizations of current algebras

We construct the explicit free field representations of the current algebras $so(2n)_k$, $so(2n+1)_k$ and $sp(2n)_k$ for a generic positive integer $n$ and an arbitrary level $k$. The corresponding energy-momentum tensors and screening currents of the first kind are also given in terms of free fields.Comment: Latex file, 25 page

Partial Deposit Insurance and Moral Hazard in Banking

Abstract: Countries with deposit insurances differ significantly on how much protection their insurance provides. We study the optimal coverage limit in a model of deposit insurance with capital requirements and risk sensitive premia to prevent moral hazard. Depositors have incentives to monitor the bank’s risk taking behavior, thus threatening banks with withdrawals of deposits if necessary. We find that either banking regulations or market discipline is insufficient to reduce bank’s risk. In addition, our numerical example explains the differences in coverage cross countries which agrees with empirical evidence. We show that low income countries provide more generous insurance protection than higher income countries.Depositor’s monitoring; moral hazard; optimal coverage, partial deposit insurance.

Multiple reference states and complete spectrum of the ZnZ_n Belavin model with open boundaries

The multiple reference state structure of the $\Z_n$ Belavin model with non-diagonal boundary terms is discovered. It is found that there exist $n$ reference states, each of them yields a set of eigenvalues and Bethe Ansatz equations of the transfer matrix. These $n$ sets of eigenvalues together constitute the complete spectrum of the model. In the quasi-classic limit, they give the complete spectrum of the corresponding Gaudin model.Comment: Latex file, 24 page

Convergence to global equilibrium for Fokker-Planck equations on a graph and Talagrand-type inequalities

In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, which is a Gibbs distribution. In this paper we study the {\em speed of convergence} towards global equilibrium for the solution of these Fokker-Planck equations on a graph, and prove that the convergence is indeed exponential. The rate as measured by the decay of the $L_2$ norm can be bound in terms of the spectral gap of the Laplacian of the graph, and as measured by the decay of (relative) entropy be bound using the modified logarithmic Sobolev constant of the graph. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [CHLZ] The first one is a local inequality, while the second is a global inequality with respect to the "lower bound metric" from [CHLZ]

Q-operator and T-Q relation from the fusion hierarchy

We propose that the Baxter $Q$-operator for the spin-1/2 XXZ quantum spin chain is given by the $j\to \infty$ limit of the transfer matrix with spin-$j$ (i.e., $(2j+1)$-dimensional) auxiliary space. Applying this observation to the open chain with general (nondiagonal) integrable boundary terms, we obtain from the fusion hierarchy the $T$-$Q$ relation for {\it generic} values (i.e. not roots of unity) of the bulk anisotropy parameter. We use this relation to determine the Bethe Ansatz solution of the eigenvalues of the fundamental transfer matrix. This approach is complementary to the one used recently to solve the same model for the roots of unity case.Comment: Latex file, 12 pages; V2, misprints corrected and references adde

Determinant Representation of Correlation Functions for the Uq(gl(1∣1))U_q(gl(1|1)) Free Fermion Model

With the help of the factorizing $F$-matrix, the scalar products of the $U_q(gl(1|1))$ free fermion model are represented by determinants. By means of these results, we obtain the determinant representations of correlation functions of the model.Comment: Latex File, 20 pages, V.3: some discussions are added, V.4 Reference update, this version will appear in J. Math. Phy