313 research outputs found

    A general approach to the sign problem - the factorization method with multiple observables

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    The sign problem is a notorious problem, which occurs in Monte Carlo simulations of a system with the partition function whose integrand is not real positive. The basic idea of the factorization method applied on such a system is to control some observables in order to determine and sample efficiently the region of configuration space which gives important contribution to the partition function. We argue that it is crucial to choose appropriately the set of the observables to be controlled in order for the method to work successfully in a general system. This is demonstrated by an explicit example, in which it turns out to be necessary to control more than one observables. Extrapolation to large system size is possible due to the nice scaling properties of the factorized functions, and known results obtained by an analytic method are shown to be consistently reproduced.Comment: 6 pages, 3 figures, (v2) references added (v3) Sections IV, V and VI improved, final version accepted by PR

    A Study of the Complex Action Problem in a Simple Model for Dynamical Compactification in Superstring Theory Using the Factorization Method

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    The IIB matrix model proposes a mechanism for dynamically generating four dimensional space--time in string theory by spontaneous breaking of the ten dimensional rotational symmetry SO(10)\textrm{SO}(10). Calculations using the Gaussian expansion method (GEM) lend support to this conjecture. We study a simple SO(4)\textrm{SO}(4) invariant matrix model using Monte Carlo simulations and we confirm that its rotational symmetry breaks down, showing that lower dimensional configurations dominate the path integral. The model has a strong complex action problem and the calculations were made possible by the use of the factorization method on the density of states ρn(x)\rho_n(x) of properly normalized eigenvalues λ~n\tilde\lambda_n of the space--time moment of inertia tensor. We study scaling properties of the factorized terms of ρn(x)\rho_n(x) and we find them in agreement with simple scaling arguments. These can be used in the finite size scaling extrapolation and in the study of the region of configuration space obscured by the large fluctuations of the phase. The computed values of λ~n\tilde\lambda_n are in reasonable agreement with GEM calculations and a numerical method for comparing the free energy of the corresponding ansatze is proposed and tested.Comment: 7 pages, 4 figures, Talk presented at the XXVIII International Symposium on Lattice Field Theory, Lattice2010, Villasimius, Italy, June 201
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