On the Correct Convergence of Complex Langevin Simulations for Polynomial Actions

There are problems in physics and particularly in field theory which are defined by complex valued weight functions $e^{-S}$ where $S$ is a polynomial action $S: R^n \rightarrow C$. The conditions under which a convergent complex Langevin calculation correctly simulates such integrals are discussed. All conditions on the process which are used to prove proper convergence are defined in the stationary limit.Comment: 8 pages, LaTeX file, preprint UNIGRAZ-UTP 29-09-9

Exact solution (by algebraic methods) of the lattice Schwinger model in the strong-coupling regime

Using the monomer--dimer representation of the lattice Schwinger model, with $N_f =1$ Wilson fermions in the strong--coupling regime ($\beta=0$), we evaluate its partition function, $Z$, exactly on finite lattices. By studying the zeroes of $Z(k)$ in the complex plane $(Re(k),Im(k))$ for a large number of small lattices, we find the zeroes closest to the real axis for infinite stripes in temporal direction and spatial extent $S=2$ and 3. We find evidence for the existence of a critical value for the hopping parameter in the thermodynamic limit $S\rightarrow \infty$ on the real axis at about $k_c \simeq 0.39$. By looking at the behaviour of quantities, such as the chiral condensate, the chiral susceptibility and the third derivative of $Z$ with respect to $1/2k$, close to the critical point $k_c$, we find some indications for a continuous phase transition.Comment: 22 pages (6 figures

Existence of positive representations for complex weights

The necessity of computing integrals with complex weights over manifolds with a large number of dimensions, e.g., in some field theoretical settings, poses a problem for the use of Monte Carlo techniques. Here it is shown that very general complex weight functions P(x) on R^d can be represented by real and positive weights p(z) on C^d, in the sense that for any observable f, _P = _p, f(z) being the analytical extension of f(x). The construction is extended to arbitrary compact Lie groups.Comment: 9 pages, no figures. To appear in J.Phys.