Exact equivalences and phase discrepancies between random matrix ensembles

We study two types of random matrix ensembles that emerge when considering the same probability measure on partitions. One is the Meixner ensemble with a hard wall and the other are two families of unitary matrix models, with weight functions that can be interpreted as characteristic polynomial insertions. We show that the models, while having the same exact evaluation for fixed values of the parameter, may present a different phase structure. We find phase transitions of the second and third order, depending on the model. Other relationships, via direct mapping, between the unitary matrix models and continuous random matrix ensembles on the real line, of Cauchy-Romanovski type, are presented and studied both exactly and asymptotically. The case of orthogonal and symplectic groups is studied as well and related to Wronskians of Chebyshev polynomials, that we evaluate at largeN.info:eu-repo/semantics/acceptedVersio

Brownian motion, Chern-Simons theory, and 2d Yang-Mills

We point out a precise connection between Brownian motion, Chern-Simons theory on S^3, and 2d Yang-Mills theory on the cylinder. The probability of reunion for N vicious walkers on a line gives the partition function of Chern-Simons theory on S^3 with gauge group U(N). The probability of starting with an equal-spacing condition and ending up with a generic configuration of movers gives the expectation value of the unknot. The probability with arbitrary initial and final states corresponds to the expectation value of the Hopf link. We find that the matrix model calculation of the partition function is nothing but the additivity law of probabilities. We establish a correspondence between quantities in Brownian motion and the modular S- and T-matrices of the WZW model at finite k and N. Brownian motion probabilitites in the affine chamber of a Lie group are shown to be related to the partition function of 2d Yang-Mills on the cylinder. Finally, the random-turns model of discrete random walks is related to Wilson's plaquette model of 2d QCD, and the latter provides an exact two-dimensional analog of the melting crystal corner. Brownian motion provides a useful unifying framework for understanding various low-dimensional gauge theories

Discrete and oscillatory matrix models in Chern-Simons theory

We derive discrete and oscillatory Chern-Simons matrix models. The method is based on fundamental properties of the associated orthogonal polynomials. As an application, we show that the discrete model allows to prove and extend the recently found relationship between Chern-Simons theory and q-deformed 2dYM. In addition, the equivalence of the Chern-Simons matrix models gives a complementary view on the equivalence of effective superpotentials in N=1 gauge theories

Chern-Simons theory, exactly solvable models and free fermions at finite temperature

We show that matrix models in Chern-Simons theory admit an interpretation as 1D exactly solvable models, paralleling the relationship between the Gaussian matrix model and the Calogero model. We compute the corresponding Hamiltonians, ground-state wavefunctions and ground-state energies and point out that the models can be interpreted as quasi-1D Coulomb plasmas. We also study the relationship between Chern-Simons theory on $S^3$ and a system of N one-dimensional fermions at finite temperature with harmonic confinement. In particular we show that the Chern-Simons partition function can be described by the density matrix of the free fermions in a very particular, crystalline, configuration. For this, we both use the Brownian motion and the matrix model description of Chern-Simons theory and find several common features with c=1 theory at finite temperature. Finally, using the exactly solvable model result, we show that the finite temperature effect can be described with a specific two-body interaction term in the Hamiltonian, with 1D Coulombic behavior at large separations.Comment: 19 pages, v2: references adde

Soft matrix models and Chern-Simons partition functions

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern-Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes-Wigert polynomials), and the deformation parameter turns out to be the usual $q$ parameter in Chern-Simons theory. In this way, we give a matrix model computation of the Chern-Simons partition function on $S^{3}$ and show that there are infinitely many matrix models with this partition function.Comment: 13 pages, 3 figure

Unitary Chern-Simons matrix model and the Villain lattice action

We use the Villain approximation to show that the Gross-Witten model, in the weak- and strong-coupling limits, is related to the unitary matrix model that describes U(N) Chern-Simons theory on S^3. The weak-coupling limit corresponds to the q->1 limit of the Chern-Simons theory while the strong-coupling regime is related to the q->0 limit. In the latter case, there is a logarithmic relationship between the respective coupling constants. We also show how the Chern-Simons matrix model arises by considering two-dimensional Yang-Mills theory with the Villain action. This leads to a U(1)^N theory which is the Abelianization of 2d Yang-Mills theory with the heat-kernel lattice action. In addition, we show that the character expansion of the Villain lattice action gives the q deformation of the heat kernel as it appears in q-deformed 2d Yang-Mills theory. We also study the relationship between the unitary and Hermitian Chern-Simons matrix models and the rotation of the integration contour in the corresponding integrals.Comment: 17 pages, Minor corrections to match the published versio

Multiplicative anomaly and zeta factorization

Some aspects of the multiplicative anomaly of zeta determinants are investigated. A rather simple approach is adopted and, in particular, the question of zeta function factorization, together with its possible relation with the multiplicative anomaly issue is discussed. We look primordially into the zeta functions instead of the determinants themselves, as was done in previous work. That provides a supplementary view, regarding the appearance of the multiplicative anomaly. Finally, we briefly discuss determinants of zeta functions that are not in the pseudodifferential operator framework.Comment: 20 pages, AIP styl