146 research outputs found
A factorization algorithm to compute Pfaffians
We describe an explicit algorithm to factorize an even antisymmetric N^2
matrix into triangular and trivial factors. This allows for a straight forward
computation of Pfaffians (including their signs) at the cost of N^3/3 flops.Comment: 6 pages, 1 figure, V2: Minor changes in the text and refs. added, to
appear in CP
Exact Algorithm for Sampling the 2D Ising Spin Glass
A sampling algorithm is presented that generates spin glass configurations of
the 2D Edwards-Anderson Ising spin glass at finite temperature, with
probabilities proportional to their Boltzmann weights. Such an algorithm
overcomes the slow dynamics of direct simulation and can be used to study
long-range correlation functions and coarse-grained dynamics. The algorithm
uses a correspondence between spin configurations on a regular lattice and
dimer (edge) coverings of a related graph: Wilson's algorithm [D. B. Wilson,
Proc. 8th Symp. Discrete Algorithms 258, (1997)] for sampling dimer coverings
on a planar lattice is adapted to generate samplings for the dimer problem
corresponding to both planar and toroidal spin glass samples. This algorithm is
recursive: it computes probabilities for spins along a "separator" that divides
the sample in half. Given the spins on the separator, sample configurations for
the two separated halves are generated by further division and assignment. The
algorithm is simplified by using Pfaffian elimination, rather than Gaussian
elimination, for sampling dimer configurations. For n spins and given floating
point precision, the algorithm has an asymptotic run-time of O(n^{3/2}); it is
found that the required precision scales as inverse temperature and grows only
slowly with system size. Sample applications and benchmarking results are
presented for samples of size up to n=128^2, with fixed and periodic boundary
conditions.Comment: 18 pages, 10 figures, 1 table; minor clarification
Pfaffians and Representations of the Symmetric Group
Pfaffians of matrices with entries z[i,j]/(x\_i+x\_j), or determinants of
matrices with entries z[i,j]/(x\_i-x\_j), where the antisymmetrical
indeterminates z[i,j] satisfy the Pl\"ucker relations, can be identified with a
trace in an irreducible representation of a product of two symmetric groups.
Using Young's orthogonal bases, one can write explicit expressions of such
Pfaffians and determinants, and recover in particular the evaluation of
Pfaffians which appeared in the recent literature.Comment: 28
Numeric and symbolic evaluation of the pfaffian of general skew-symmetric matrices
Evaluation of pfaffians arises in a number of physics applications, and for
some of them a direct method is preferable to using the determinantal formula.
We discuss two methods for the numerical evaluation of pfaffians. The first is
tridiagonalization based on Householder transformations. The main advantage of
this method is its numerical stability that makes unnecessary the
implementation of a pivoting strategy. The second method considered is based on
Aitken's block diagonalization formula. It yields to a kind of LU (similar to
Cholesky's factorization) decomposition (under congruence) of arbitrary
skew-symmetric matrices that is well suited both for the numeric and symbolic
evaluations of the pfaffian. Fortran subroutines (FORTRAN 77 and 90)
implementing both methods are given. We also provide simple implementations in
Python and Mathematica for purpose of testing, or for exploratory studies of
methods that make use of pfaffians.Comment: 13 pages, Download links:
http://gamma.ft.uam.es/robledo/Downloads.html and
http://www.phys.washington.edu/users/bertsch/computer.htm
The Pfaff lattice and skew-orthogonal polynomials
Consider a semi-infinite skew-symmetric moment matrix, m_{\iy} evolving
according to the vector fields \pl m / \pl t_k=\Lb^k m+m \Lb^{\top k} , where
\Lb is the shift matrix. Then the skew-Borel decomposition m_{\iy}:= Q^{-1}
J Q^{\top -1} leads to the so-called Pfaff Lattice, which is integrable, by
virtue of the AKS theorem, for a splitting involving the affine symplectic
algebra. The tau-functions for the system are shown to be pfaffians and the
wave vectors skew-orthogonal polynomials; we give their explicit form in terms
of moments. This system plays an important role in symmetric and symplectic
matrix models and in the theory of random matrices (beta=1 or 4).Comment: 21 page
The invariants of a genus one curve
It was first pointed out by Weil that we can use classical invariant theory
to compute the Jacobian of a genus one curve. The invariants required for
curves of degree n = 2,3,4 were already known to the nineteenth centuary
invariant theorists. We have succeeded in extending these methods to curves of
degree n = 5, where although the invariants are too large to write down as
explicit polynomials, we have found a practical algorithm for evaluating them.Comment: 37 page
- …