On the spectrum of stiffness matrices arising from isogeometric analysis

We study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems. Motivated by the applicative interest in the fast solution of the related linear systems, we are looking for a spectral characterization of the involved matrices. In particular, we investigate non-singularity, conditioning (extremal behavior), spectral distribution in the Weyl sense, as well as clustering of the eigenvalues to a certain (compact) subset of \(\mathbb C\) . All the analysis is related to the notion of symbol in the Toeplitz setting and is carried out both for the cases of 1D and 2D problems.     

Critical speeding-up in the local dynamics of the random-cluster model

We study the dynamic critical behavior of the local bond-update (Sweeny) dynamics for the Fortuin-Kasteleyn random-cluster model in dimensions d=2, 3 by Monte Carlo simulation. We show that, for a suitable range of q values, the global observable S2 exhibits "critical speeding-up": it decorrelates well on time scales much less than one sweep. In some cases the dynamic critical exponent for the integrated autocorrelation time is negative. We also show that the dynamic critical exponent zexp is very close (possibly equal) to the rigorous lower bound alpha/nu and quite possibly smaller than the corresponding exponent for the Chayes-Machta-Swendsen-Wang cluster dynamics.     

A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices

Numerical Algorithms - In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained the precise asymptotic expansion for the eigenvalues of a Toeplitz matrix Tn(f), under suitable...     

Critical behavior of the Chayes-Machta-Swendsen-Wang dynamics

We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z >or= alpha/nu is close to but probably not sharp in d = 2 and is far from sharp in d = 3, for all q. The conjecture z >or= beta/nu is false (for some values of q) in both d = 2 and d = 3.     

A worm algorithm for the fully-packed loop model

We present a Markov-chain Monte Carlo algorithm of worm   type that correctly simulates the fully-packed loop model with n=1$n=1$ on the honeycomb lattice, and we prove that it is ergodic and has uniform stationary distribution. The honeycomb-lattice fully-packed loop model with n=1$n=1$ is equivalent to the zero-temperature triangular-lattice antiferromagnetic Ising model, which is fully frustrated and notoriously difficult to simulate. We test this worm algorithm numerically and estimate the dynamic exponent zexp=0.515(8)$z_{\exp }=0.515(8)$. We also measure several static quantities of interest, including loop-length and face-size moments. It appears numerically that the face-size moments are governed by the magnetic dimension for percolation.     

On the Coupling Time of the Heat-Bath Process for the Fortuin–Kasteleyn Random–Cluster Model

We consider the coupling from the past implementation of the random–cluster heat-bath process, and study its random running time, or coupling time. We focus on hypercubic lattices embedded on tori, in dimensions one to three, with cluster fugacity at least one. We make a number of conjectures regarding the asymptotic behaviour of the coupling time, motivated by rigorous results in one dimension and Monte Carlo simulations in dimensions two and three. Amongst our findings, we observe that, for generic parameter values, the distribution of the appropriately standardized coupling time converges to a Gumbel distribution, and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time. Perhaps surprisingly, we observe these results to hold both off criticality, where the coupling time closely mimics the coupon collector’s problem, and also at the critical point, provided the cluster fugacity is below the value at which the transition becomes discontinuous. Finally, we consider analogous questions for the single-spin Ising heat-bath process.     

Dynamic Critical Behavior of the Chayes–Machta Algorithm for the Random-Cluster Model,I. Two Dimensions

We study, via Monte Carlo simulation, the dynamic critical behavior of the Chayes–Machta dynamics for the Fortuin–Kasteleyn random-cluster model, which generalizes the Swendsen–Wang dynamics for the q-state Potts ferromagnet to non-integer q≥1. We consider spatial dimension d=2 and 1.25≤q≤4 in steps of 0.25, on lattices up to 1024², and obtain estimates for the dynamic critical exponent z _CM. We present evidence that when 1≤q≲1.95 the Ossola–Sokal conjecture z _CM≥β/ν is violated, though we also present plausible fits compatible with this conjecture. We show that the Li–Sokal bound z _CM≥α/ν is close to being sharp over the entire range 1≤q≤4, but is probably non-sharp by a power. As a byproduct of our work, we also obtain evidence concerning the corrections to scaling in static observables.     

Worm Monte Carlo study of the honeycomb-lattice loop model

We present a Markov-chain Monte Carlo algorithm of worm   type that correctly simulates the O(n)$O(n)$ loop model on any (finite and connected) bipartite cubic graph, for any real n>0$n>0$, and any edge weight, including the fully-packed limit of infinite edge weight. Furthermore, we prove rigorously that the algorithm is ergodic and has the correct stationary distribution. We emphasize that by using known exact mappings when n=2$n=2$, this algorithm can be used to simulate a number of zero-temperature Potts antiferromagnets for which the Wang–Swendsen–Kotecký cluster algorithm is non-ergodic, including the 3-state model on the kagome lattice and the 4-state model on the triangular lattice. We then use this worm algorithm to perform a systematic study of the honeycomb-lattice loop model as a function of n?2$n?2$, on the critical line and in the densely-packed and fully-packed phases. By comparing our numerical results with Coulomb gas theory, we identify a set of exact expressions for scaling exponents governing some fundamental geometric and dynamic observables. In particular, we show that for all n?2$n?2$, the scaling of a certain return time in the worm dynamics is governed by the magnetic dimension of the loop model, thus providing a concrete dynamical interpretation of this exponent. The case n>2$n>2$ is also considered, and we confirm the existence of a phase transition in the 3-state Potts universality class that was recently observed via numerical transfer matrix calculations.     

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

 The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in Σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations. Received: 24 July 2002 / Accepted: 26 January 2003 Published online: 13 May 2003 Communicated by L. Takhtajan