Improving Random Number Generators in the Monte Carlo simulations via twisting and combining

Problems for various random number generators accompanying the Wolff algorithm [U. Wolff, Phys. Rev. Lett. 62 (1989) 361; U. Wolff, Phys. Lett. B 228 (1989) 379] are discussed, including the hidden errors first reported in [A.M. Ferrenberg, D.P. Landau, Y.J. Wong, Phys. Rev. Lett. 69 (1992) 3382]. A general (though simple) method of twisting and combining for improving the performance of these generators is proposed. Some recent generators motivated by such a twisting and combining method with extremely long period are discussed. The proposed method provides a novel and simple way to improve RNGs in its performance.

[1]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[2]  R. Tausworthe.  Random Numbers Generated by Linear Recurrence Modulo Two , 1965 .

[3]  Pierre L'Ecuyer,et al.  Bad Lattice Structures for Vectors of Nonsuccessive Values Produced by Some Linear Recurrences , 1997, INFORMS J. Comput..

[4]  Peter Grassberger,et al.  On correlations in “good” random number generators , 1993 .

[5]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: List of Symbols , 1986 .

[6]  Andrew G. Glen,et al.  APPL , 2001 .

[7]  C. Pomerance,et al.  Prime Numbers: A Computational Perspective , 2002 .

[8]  Stephan Mertens,et al.  Entropy of pseudo-random-number generators. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Pierre L'Ecuyer,et al.  Beware of linear congruential generators with multipliers of the form a = ±2q ±2r , 1999, TOMS.

[10]  G. Marsaglia,et al.  A New Class of Random Number Generators , 1991 .

[11]  Nikos L. Doltsinis,et al.  Quantum Simulations of complex many-body systems:from theory to algorithms , 2002 .

[12]  Lih-Yuan Deng,et al.  A system of high-dimensional, efficient, long-cycle and portable uniform random number generators , 2003, TOMC.

[13]  Mei Ling Huang.  Recurrence relations for the r-distribution , 1990 .

[14]  B. M. Fulk.  MATH , 1992 .

[15]  S. Papson.  “Model” , 1981 .

[16]  Wolff,et al.  Collective Monte Carlo updating for spin systems. , 1989, Physical review letters.

[17]  Scott Kirkpatrick,et al.  A very fast shift-register sequence random number generatorjournal of computational physics , 1981 .

[18]  Alan M. Ferrenberg,et al.  Monte Carlo simulations: Hidden errors from "good" random number generators. , 1992, Physical review letters.

[19]  M. Fisher,et al.  Bounded and Inhomogeneous Ising Models. I. Specific-Heat Anomaly of a Finite Lattice , 1969 .

[20]  Pierre L'Ecuyer,et al.  TestU01: A C library for empirical testing of random number generators , 2006, TOMS.

[21]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[22]  Ted G. Lewis,et al.  Generalized Feedback Shift Register Pseudorandom Number Algorithm , 1973, JACM.

[23]  Beale.  Exact distribution of energies in the two-dimensional ising model. , 1996, Physical review letters.

[24]  Vattulainen,et al.  Physical tests for random numbers in simulations. , 1994, Physical review letters.

[25]  Craig B. Borkowf,et al.  Random Number Generation and Monte Carlo Methods , 2000, Technometrics.

[26]  U. Wolff.  Comparison Between Cluster Monte Carlo Algorithms in the Ising Model , 1989 .

[27]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[28]  N. B. Wilding,et al.  Errors In Monte Carlo Simulations Using Shift Register Random Number Generators , 1995 .

[29]  I. D. Hill,et al.  An Efficient and Portable Pseudo‐Random Number Generator , 1982 .

[30]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[31]  F. James,et al.  RANLUX: A Fortran implementation of the high-quality pseudorandom number generator of Lüscher , 1994 .

[32]  Shlomo Levental,et al.  On pricing of market-indexed certificates of deposit , 1989 .

[33]  Lih-Yuan Deng,et al.  Efficient and portable multiple recursive generators of large order , 2005, TOMC.

[34]  Lih-Yuan Deng,et al.  64-Bit and 128-bit DX random number generators , 2010, Computing.

[35]  M. Luescher,et al.  A Portable High-quality Random Number Generator for Lattice Field Theory Simulations , 1993 .

[36]  T. Ala‐Nissila,et al.  Physical models as tests of randomness. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[37]  Giovanni Ossola,et al.  Systematic errors due to linear congruential random-number generators with the Swendsen-Wang algorithm: a warning. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Lev N. Shchur,et al.  Cluster Monte Carlo: tScaling of systematic errors in the two-dimensional Ising model , 1997 .

[39]  J. R. Heringa,et al.  Simulation of a directed random-walk model the effect of pseudo-random-number correlations , 1996, cond-mat/9612227.

[40]  P. D. Coddington,et al.  Analysis of random number generators using Monte Carlo simulation , 1993, cond-mat/9309017.

[41]  George Marsaglia,et al.  A random number generator for PC's , 1990 .

[42]  Dennis K. J. Lin,et al.  Random Number Generation for the New Century , 2000 .