A Monte Carlo algorithm for sampling rare events: application to a search for the Griffiths singularity

In and beyond the Griffiths phase: A large-deviation study of the magnetic susceptibility of the two-dimensional bond-diluted Ising model

The Griffiths phase in systems with quenched disorder occurs below the ordering transition of the pure system down to the ordering transition of the actual disordered system. While it does not exhibit long-range order, large fluctuations in the disorder degrees of freedom result in exponentially rare, long-range ordered states and hence the occurrence of broad distributions in response functions. Inside the Griffiths phase of the two-dimensional bond-diluted Ising model the distribution of the magnetic susceptibility is expected to have such a broad, exponential tail. A large-deviation Monte Carlo algorithm is used to sample this distribution and the exponential tail is extracted over a wide range of the support down to very small probabilities of the order of 10^{-300}. We study the behavior of the susceptibility distribution across the full phase diagram, from the paramagnetic state through the Griffiths phase to the ferromagnetically ordered system and down to the zero-temperature point. We extract the rate function of large-deviation theory as well as its finite-size scaling behavior and we reveal interesting differences and similarities between the cases. A connection between the fraction of ferromagnetic bonds in a given disorder sample and the size of the magnetic susceptibility is demonstrated numerically.

Multicanonical sampling of rare events in random matrices

A method based on multicanonical Monte Carlo is applied to the calculation of large deviations in the largest eigenvalue of random matrices. The method is successfully tested with the Gaussian orthogonal ensemble, sparse random matrices, and matrices whose components are subject to uniform density. Specifically, the probability that all eigenvalues of a matrix are negative is estimated in these cases down to the values of ∼10(-200), a region where simple random sampling is ineffective. The method can be applied to any ensemble of matrices and used for sampling rare events characterized by any statistics.

Minimum-free-energy distribution of RNA secondary structures: Entropic and thermodynamic properties of rare events

We study the distribution of the minimum free energy (MFE) for the Turner model of pseudoknot free RNA secondary structures over ensembles of random RNA sequences. In particular, we are interested in those rare and intermediate events of unexpected low MFEs. Generalized ensemble Markov-chain Monte Carlo methods allow us to explore the rare-event tail of the MFE distribution down to probabilities such as 10^{-70} and to study the relationship between the sequence entropy and structural properties for sequence ensembles with fixed MFEs. Entropic and structural properties of those ensembles are compared with natural RNA of the same reduced MFE ( z score).