© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

DEVELOPING A TEST FOR RANDOM NUMBER GENERATORS USING A SIMULATION OF THE HIERARCHICAL POTTS DIAMOND MODEL AT THE CRITICAL POINT

Joshua Liberty

Published Jan 1, 2014• 0 citations

Abstract

This dissertation uses the hierarchical q-state Potts model at the critical point to develop a new random number generator test. We start with an exposition of renormalization group approach by means of which one can numerically exactly compute the free energy, specific heat and susceptibility of large, but finite lattices. We then show that generalization of these standard techniques allows one to also compute probability distributions related to the energy and the order parameter. The various computed quantities can be compared with Monte Carlo estimates of the same quantities. We demonstrate that the structure of the hierarchical lattices used allows one to perform the Monte Carlo calculations by direct sampling. This avoids the usual critical slowing down that plagues Monte Carlo calculations at the critical point. As is well known, critical behavior is highly susceptible to perturbations. We expect that flaws of the pseudo random number generator, such as correlations, will cause statistically significant discrepancies between the results of the simulations and the numerically exactly computed results. Details of the computer code generated for these tests are included.

DOI: 10.23860/diss-liberty-joshua-2014

Related papers

Monte Carlo algorithm for the random cluster model

Christian R. Scullard

I present a generalization of the Newman-Ziff algorithm, a standard technique in numerical percolation, to the random cluster model. The new method is straightfoward to implement, works for real $q>0$, and is completely free of the critical slowdown problem. By using the method to sample the critical polynomial, accurate estimates of Potts critical points can be found using relatively small lattice sizes. Furthermore, results for all values of $q$ can be found at once within a single simulation. I demonstrate its utility here by computing critical points for non-integer values of $q$ on the square lattice to compare with the exact solution, and on the non-planar square matching lattice.

arXiv:2102.12716•Feb 25, 2021•0 citations

Critical point of the Kagomé Potts model: a Monte Carlo renormalization group and scaling determination

Jau-Ann Chen, Chin‐Kun Hu, Fa Yueh Wu

The determination of the critical point of the q -state Potts model on the Kagomé lattices has been an outstanding unsolved problem. Here we study this problem by regarding the Potts model as generating correlated bond percolations. By applying a combination of Monte Carlo renormalization group and finite-size scaling analyses to the percolation problem, numerical estimates with an accuracy of are obtained for the Kagomé critical point for q = 1,2,3,4. Our results for q = 1 are consistent with a recent highly accurate numerical estimate by Ziff and Suding, and results for q = 2 agree with the known exact result within the numerical accuracy. Compared with results obtained in a recent series analysis by Jensen et al , our numbers differ from theirs slightly for q = 3, and agree with theirs with a slightly better accuracy for q = 4. Our numbers also confirm that a conjecture due to Wu is extremely accurate.

Journal of Physics A Mathematical and General•Oct 2, 1998•11 citations

Disordered Potts model on the diamond hierarchical lattice: Numerically exact treatment in the large- q limit

Ferenc Iglói, Loïc Turban

We consider the critical behavior of the random $q$-state Potts model in the large-$q$ limit with different types of disorder leading to either the nonfrustrated random ferromagnet regime or the frustrated spin-glass regime. The model is studied on the diamond hierarchical lattice for which the Migdal-Kadanoff real-space renormalization is exact. It is shown to have a ferromagnetic and a paramagnetic phase and the phase transition is controlled by four different fixed points. The state of the system is characterized by the distribution of the interface free energy $P(I)$ which is shown to satisfy different integral equations at the fixed points. By numerical integration we have obtained the corresponding stable laws of nonlinear combination of random numbers and obtained numerically exact values for the critical exponents.

arXiv:0908.3622•Physical Review B•Oct 9, 2009•11 citations

Comparing pseudo- and quantum-random number generators with Monte Carlo simulations

David Cirauqui, Miguel Ángel García-March, Guillem Guigó i Corominas +5 more

We study how the Monte Carlo simulations of the critical dynamics of two-dimensional Ising lattices are affected by the quality (as compared to true randomness) of the pseudo-randomness used in them by computing an observable known to be sensitive to it: the dynamic critical exponent z. To this end, we first present a review of the literature and then perform the same simulations with different pseudo-randomness sources. We control the quality of the random streams by physically reseeding the generators and observe that while the mean value of the measured observable is slightly affected by the correlations within them, its variance does get a severe impact. By studying this affectation and how such pseudo-random streams perform under standard randomness tests based on statistical analysis, we conclude that the new protocol is able to detect other types of correlations and can thus be used as an additional test. On the other hand, we present the first (to the best of our knowledge) computation of the dynamic critical exponent with a perfectly random quantum random number generator (QRNG) and hypothesize that the presented strategy may point toward a route to establish quantum advantage based on the quality of results as complementary to those strategies based on the speed of the computational task.

APL Quantum•Sep 1, 2024•5 citations

The Random Field Ising Model on Hierarchical Lattices II: Ground State Critical Properties

Alexandre Rosas, S. Coutinho

The ground state critical properties of the Random Field Ising Model (RFIM) on the diamond hierarchical lattice are investigated via a combining method encompassing real space renormalization group and an exact recurrence procedure. The local magnetization and the nearest neighbors pair correlation function are exactly calculated. The fixed-point joint probability distribution of couplings and local fields are numerically obtained and analyzed, indicating that the critical behavior of the model is governed by the zero temperature disorder fixed point. The critical exponents associated with the order parameter and correlation length are estimated showing an universal behavior regarding the choice of the initial probability distribution for initial local fields being the symmetric continuous Gaussian or discrete delta-bimodal.

arXiv:cond-mat/0107407•Jul 19, 2001•0 citations

CRITICAL POINT OF THE KAGOME POTTS MODEL: A HISTOGRAM MONTE CARLO RENORMALIZATION GROUP DETERMINATION

Chin‐Kun Hu, Jau-Ann Chen, Fa Yueh Wu

The histogram Monte Carlo renormalization group method proposed by Hu is used to determine the critical point of the q-state Potts model on the Kagome lattice. Our results are compared with the predictions of conjectures by Wu and Tsallis.

Modern Physics Letters B•Mar 20, 1994•6 citations

This dissertation uses the hierarchical q-state Potts model at the critical point to develop a new random number generator test. We start with an exposition of renormalization group approach by means of which one can numerically exactly compute the free energy, specific heat and susceptibility of large, but finite lattices. We then show that generalization of these standard techniques allows one to also compute probability distributions related to the energy and the order parameter. The various computed quantities can be compared with Monte Carlo estimates of the same quantities. We demonstrate that the structure of the hierarchical lattices used allows one to perform the Monte Carlo calculations by direct sampling. This avoids the usual critical slowing down that plagues Monte Carlo calculations at the critical point. As is well known, critical behavior is highly susceptible to perturbations. We expect that flaws of the pseudo random number generator, such as correlations, will cause statistically significant discrepancies between the results of the simulations and the numerically exactly computed results. Details of the computer code generated for these tests are included.

I present a generalization of the Newman-Ziff algorithm, a standard technique in numerical percolation, to the random cluster model. The new method is straightfoward to implement, works for real

q>0

, and is completely free of the critical slowdown problem. By using the method to sample the critical polynomial, accurate estimates of Potts critical points can be found using relatively small lattice sizes. Furthermore, results for all values of

q

can be found at once within a single simulation. I demonstrate its utility here by computing critical points for non-integer values of

q

on the square lattice to compare with the exact solution, and on the non-planar square matching lattice.

The determination of the critical point of the q -state Potts model on the Kagomé lattices has been an outstanding unsolved problem. Here we study this problem by regarding the Potts model as generating correlated bond percolations. By applying a combination of Monte Carlo renormalization group and finite-size scaling analyses to the percolation problem, numerical estimates with an accuracy of are obtained for the Kagomé critical point for q = 1,2,3,4. Our results for q = 1 are consistent with a recent highly accurate numerical estimate by Ziff and Suding, and results for q = 2 agree with the known exact result within the numerical accuracy. Compared with results obtained in a recent series analysis by Jensen et al , our numbers differ from theirs slightly for q = 3, and agree with theirs with a slightly better accuracy for q = 4. Our numbers also confirm that a conjecture due to Wu is extremely accurate.

We consider the critical behavior of the random

q

-state Potts model in the large-

q

limit with different types of disorder leading to either the nonfrustrated random ferromagnet regime or the frustrated spin-glass regime. The model is studied on the diamond hierarchical lattice for which the Migdal-Kadanoff real-space renormalization is exact. It is shown to have a ferromagnetic and a paramagnetic phase and the phase transition is controlled by four different fixed points. The state of the system is characterized by the distribution of the interface free energy

P(I)

which is shown to satisfy different integral equations at the fixed points. By numerical integration we have obtained the corresponding stable laws of nonlinear combination of random numbers and obtained numerically exact values for the critical exponents.

We study how the Monte Carlo simulations of the critical dynamics of two-dimensional Ising lattices are affected by the quality (as compared to true randomness) of the pseudo-randomness used in them by computing an observable known to be sensitive to it: the dynamic critical exponent z. To this end, we first present a review of the literature and then perform the same simulations with different pseudo-randomness sources. We control the quality of the random streams by physically reseeding the generators and observe that while the mean value of the measured observable is slightly affected by the correlations within them, its variance does get a severe impact. By studying this affectation and how such pseudo-random streams perform under standard randomness tests based on statistical analysis, we conclude that the new protocol is able to detect other types of correlations and can thus be used as an additional test. On the other hand, we present the first (to the best of our knowledge) computation of the dynamic critical exponent with a perfectly random quantum random number generator (QRNG) and hypothesize that the presented strategy may point toward a route to establish quantum advantage based on the quality of results as complementary to those strategies based on the speed of the computational task.

The ground state critical properties of the Random Field Ising Model (RFIM) on the diamond hierarchical lattice are investigated via a combining method encompassing real space renormalization group and an exact recurrence procedure. The local magnetization and the nearest neighbors pair correlation function are exactly calculated. The fixed-point joint probability distribution of couplings and local fields are numerically obtained and analyzed, indicating that the critical behavior of the model is governed by the zero temperature disorder fixed point. The critical exponents associated with the order parameter and correlation length are estimated showing an universal behavior regarding the choice of the initial probability distribution for initial local fields being the symmetric continuous Gaussian or discrete delta-bimodal.