52 research outputs found
Exact integration of height probabilities in the Abelian Sandpile Model
The height probabilities for the recurrent configurations in the Abelian
Sandpile Model on the square lattice have analytic expressions, in terms of
multidimensional quadratures. At first, these quantities have been evaluated
numerically with high accuracy, and conjectured to be certain cubic
rational-coefficient polynomials in 1/pi. Later their values have been
determined by different methods.
We revert to the direct derivation of these probabilities, by computing
analytically the corresponding integrals. Yet another time, we confirm the
predictions on the probabilities, and thus, as a corollary, the conjecture on
the average height.Comment: 17 pages, added reference
An exactly solvable random satisfiability problem
We introduce a new model for the generation of random satisfiability
problems. It is an extension of the hyper-SAT model of Ricci-Tersenghi, Weigt
and Zecchina, which is a variant of the famous K-SAT model: it is extended to
q-state variables and relates to a different choice of the statistical
ensemble. The model has an exactly solvable statistic: the critical exponents
and scaling functions of the SAT/UNSAT transition are calculable at zero
temperature, with no need of replicas, also with exact finite-size corrections.
We also introduce an exact duality of the model, and show an analogy of
thermodynamic properties with the Random Energy Model of disordered spin
systems theory. Relations with Error-Correcting Codes are also discussed.Comment: 31 pages, 1 figur
Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities II. Grassmann and quantum oscillator algebra representation
We prove that, for , , and matrices with entries in a
non-commutative ring such that ,
satisfying suitable commutation relations (in particular, is a Manin
matrix), the following identity holds: . Furthermore,
if also is a Manin matrix, . Notations: , , are respectively the bra and the ket of the ground state,
and the creation and annihilation operators of a quantum
harmonic oscillator, while and are Grassmann
variables in a Berezin integral. These results should be seen as a
generalization of the classical Cauchy-Binet formula, in which and are
null matrices, and of the non-commutative generalization, the Capelli identity,
in which and are identity matrices and
.Comment: 40 page
The complexity of the Multiple Pattern Matching Problem for random strings
We generalise a multiple string pattern matching algorithm, recently proposed
by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary
dictionaries on an alphabet of size . If is the number of words of
length in the dictionary, and , the
complexity rate for the string characters to be read by this algorithm is at
most for some constant
. On the other side, we generalise the classical lower
bound of Yao [SIAM J. Comput. 8, 1979], for the problem with a single pattern,
to deal with arbitrary dictionaries, and determine it to be at least
. This proves the optimality of the
algorithm, improving and correcting previous claims.Comment: 25 pages, 4 figure
Conservation laws for strings in the Abelian Sandpile Model
The Abelian Sandpile generates complex and beautiful patterns and seems to
display allometry. On the plane, beyond patches, patterns periodic in both
dimensions, we remark the presence of structures periodic in one dimension,
that we call strings. We classify completely their constituents in terms of
their principal periodic vector k, that we call momentum. We derive a simple
relation between the momentum of a string and its density of particles, E,
which is reminiscent of a dispersion relation, E=k^2. Strings interact: they
can merge and split and within these processes momentum is conserved. We reveal
the role of the modular group SL(2,Z) behind these laws.Comment: 4 pages, 4 figures in colo
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