We consider random non-hermitean matrices in the large N limit. The power of analytic function theory cannot be brought to bear directly to analyze non-hermitean random matrices, in contrast to hermitean random matrices. To overcome this difficulty, we show that associated to each ensemble of nonhermitean matrices there is an auxiliary ensemble of random hermitean matrices which can be analyzed by the usual methods. We then extract the Green's function and the density of eigenvalues of the non-hermitean ensemble from those of the auxiliary ensemble. We apply this "method of hermitization" to several examples, and discuss a number of related issues.

We study localization and delocalization in a class of non-hermitean Hamiltonians inspired by the problem of vortex pinning in superconductors. In various simplified models we are able to obtain analytic descriptions, in particular of the non-perturbative emergence of a forked structure (the appearance of "wings") in the density of states. We calculate how the localization length diverges at the localization-delocalization transition. We map some versions of this problem onto a random walker problem in two dimensions. For a certain model, we find an intricate structure in its density of states.

We discuss supersymmetric quantum mechanical models with periodic potentials. The important new feature is that it is possible for both isospectral potentials to support zero modes, in contrast with the standard nonperiodic case where either one or neither ͑but not both͒ of the isospectral pair has a zero mode. Thus it is possible to have supersymmetry unbroken and yet also have a vanishing Witten index. We present some explicit exactly soluble examples for which the isospectral potentials have identical band spectra, and which are ''self-isospectral'' in the sense that the potentials have identical shape, but are translated by one half-period relative to one another. ͓S0556-2821͑98͒02202-4͔ Supersymmetry and supersymmetry breaking are fundamental issues in theoretical particle physics, and supersymmetric ͑SUSY͒ quantum mechanics ͑QM͒ provides an important testing ground for both physical and computational aspects of SUSY theories ͓1,2͔. There are also many applications to the theory of solitons ͓3͔. Of particular interest for particle physics are possible mechanisms for breaking SUSY dynamically. Typically, one considers models with discrete spectra, and then the Witten index, which characterizes the difference between the number of bosonic and fermionic zero modes, may be used to indicate whether or not SUSY is broken ͓1͔. Interesting subtleties arise for potentials with continuum states ͓4͔ or with singularities ͓5͔.In this paper we consider SUSY quantum mechanics for periodic potentials ͑which therefore have band spectra͒. The main new feature is that it is possible for the periodic isospectral bosonic and fermionic potentials to have exactly the same spectrum, including zero modes. This is in contrast with the usual ͑nonperiodic and fast decaying͒ case for which at most one potential of an isospectral pair can have a zero mode.Consider one dimensional SUSY quantum mechanical models on the real line. The bosonic and fermionic Hamiltonians H Ϯ correspond to an isospectral pair of potentials

We apply the recently introduced method of hermitization to study in the large N limit non-hermitean random matrices that are drawn from a large class of circularly symmetric non-Gaussian probability distributions, thus extending the recent Gaussian non-hermitean literature. We develop the general formalism for calculating the Green's function and averaged density of eigenvalues, which may be thought of as the non-hermitean analog of the method due to Brèzin, Itzykson, Parisi and Zuber for analyzing hermitean non-Gaussian random matrices. We obtain an explicit algebraic equation for the integrated density of eigenvalues. A somewhat surprising result of that equation is that the shape of the eigenvalue distribution in the complex plane is either a disk or an annulus. As a concrete example, we analyze the quartic ensemble and study the phase transition from a disk shaped eigenvalue distribution to an annular distribution. Finally, we apply the method of hermitization to develop the addition formalism for free non-hermitean random variables. We use this formalism to state and prove a non-abelian non-hermitean version of the central limit theorem. *

We review in detail the construction of all stable static fermion bags in the 1 + 1 dimensional Gross-Neveu model with N flavors of Dirac fermions, in the large N limit. In addition to the well known kink and topologically trivial solitons (which correspond, respectively, to the spinor and antisymmetric tensor representations of O(2N )), there are also threshold bound states of a kink and a topologically trivial soliton: the heavier topological solitons (HTS). The mass of any of these newly discovered HTS's is the sum of masses of its solitonic constituents, and it corresponds to the tensor product of their O(2N ) representations. Thus, it is marginally stable (at least in the large N limit). Furthermore, its mass is independent of the distance between the centers of its constituents, which serves as a flat collective coordinate, or a modulus. There are no additional stable static solitons in the Gross-Neveu model. We provide detailed derivation of the profiles, masses and fermion number contents of these static solitons. For pedagogical clarity, and in order for this paper to be self-contained, we also included detailed appendices on supersymmetric quantum mechanics and on reflectionless potentials in one spatial dimension, which are intimately related with the theory of static fermion bags. In particular, we present a novel simple explicit formula for the diagonal resolvent of a reflectionless Schrödinger operator with an arbitrary number of bound states. In additional appendices we summarize the relevant group representation theoretic facts, and also provide a simple calculation of the mass of the kinks.