Spanning forests, electrical networks, and a determinant identity

We aim to generalize a theorem on the number of rooted spanning forests of a highly symmetric graph to the case of asymmetric graphs. We show that this can be achieved by means of an identity between the minor determinants of a Laplace matrix, for which we provide two different (combinatorial as well as algebraic) proofs in the simplest case. Furthermore, we discuss the connections to electrical networks and the enumeration of spanning trees in sequences of self-similar graphs

The two-dimensional two-component plasma plus background on a sphere : Exact results

An exact solution is given for a two-dimensional model of a Coulomb gas, more general than the previously solved ones. The system is made of a uniformly charged background, positive particles, and negative particles, on the surface of a sphere. At the special value $\Gamma = 2$ of the reduced inverse temperature, the classical equilibrium statistical mechanics is worked out~: the correlations and the grand potential are calculated. The thermodynamic limit is taken, and as it is approached the grand potential exhibits a finite-size correction of the expected universal form.Comment: 23 pages, Plain Te

The N=2\mathcal{N}=2 Schur index from free fermions

We study the Schur index of 4-dimensional $\mathcal{N}=2$ circular quiver theories. We show that the index can be expressed as a weighted sum over partition functions describing systems of free Fermions living on a circle. For circular $SU(N)$ quivers of arbitrary length we evaluate the large $N$ limit of the index, up to exponentially suppressed corrections. For the single node theory ($\mathcal{N}=4$ SYM) and the two node quiver we are able to go beyond the large $N$ limit, and obtain the complete, all orders large $N$ expansion of the index, as well as explicit finite $N$ results in terms of elliptic functions.Comment: 36 pages, 1 figure; v2: Minor corrections, version published in JHEP; v3: Minor correction

An equilibrium problem for the limiting eigenvalue distribution of rational Toeplitz matrices

We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem. Moreover, we show that the other components describe the limiting behavior of certain generalized eigenvalues. In this way, we generalize the recent results of Duits and Kuijlaars for banded Toeplitz matrices.Comment: 20 pages, 2 figure