Random Series and Discrete Path Integral methods: The Levy-Ciesielski implementation

Details Random Series and Discrete Path Integral methods: The Levy-Ciesielski implementation Random Series and Discrete Path Integral methods: The Levy-Ciesielski implementation

2002-12-10

arxiv.org/abs/cond-mat/0212219v2

Phys. Rev. E 67, 026124 (2003)

Physics

Condensed Matter

Statistical Mechanics

20 pages, 4 figures; the two equations before Eq. 14 are corrected; other typos are removed

Scientific paper

10.1103/PhysRevE.67.026124

We perform a thorough analysis of the relationship between discrete and series representation path integral methods, which are the main numerical techniques used in connection with the Feynman-Kac formula. First, a new interpretation of the so-called standard discrete path integral methods is derived by direct discretization of the Feynman-Kac formula. Second, we consider a particular random series technique based upon the Levy-Ciesielski representation of the Brownian bridge and analyze its main implementations, namely the primitive, the partial averaging, and the reweighted versions. It is shown that the n=2^k-1 subsequence of each of these methods can also be interpreted as a discrete path integral method with appropriate short-time approximations. We therefore establish a direct connection between the discrete and the random series approaches. In the end, we give sharp estimates on the rates of convergence of the partial averaging and the reweighted Levy-Ciesielski random series approach for sufficiently smooth potentials. The asymptotic rates of convergence are found to be O(1/n^2), in agreement with the rates of convergence of the best standard discrete path integral techniques.

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