38,745 research outputs found
Paths and partitions: combinatorial descriptions of the parafermionic states
The Z_k parafermionic conformal field theories, despite the relative
complexity of their modes algebra, offer the simplest context for the study of
the bases of states and their different combinatorial representations. Three
bases are known. The classic one is given by strings of the fundamental
parafermionic operators whose sequences of modes are in correspondence with
restricted partitions with parts at distance k-1 differing at least by 2.
Another basis is expressed in terms of the ordered modes of the k-1 different
parafermionic fields, which are in correspondence with the so-called multiple
partitions. Both types of partitions have a natural (Bressoud) path
representation. Finally, a third basis, formulated in terms of different paths,
is inherited from the solution of the restricted solid-on-solid model of
Andrews-Baxter-Forrester. The aim of this work is to review, in a unified and
pedagogical exposition, these four different combinatorial representations of
the states of the Z_k parafermionic models.
The first part of this article presents the different paths and partitions
and their bijective relations; it is purely combinatorial, self-contained and
elementary; it can be read independently of the conformal-field-theory
applications. The second part links this combinatorial analysis with the bases
of states of the Z_k parafermionic theories. With the prototypical example of
the parafermionic models worked out in detail, this analysis contributes to fix
some foundations for the combinatorial study of more complicated theories.
Indeed, as we briefly indicate in ending, generalized versions of both the
Bressoud and the Andrews-Baxter-Forrester paths emerge naturally in the
description of the minimal models.Comment: 53 pages (v2: minor modifications,v3: 3 typos corrected); to appear
in the special issue of J. Math. Phys. on "Integrable Quantum Systems and
Solvable Statistical Mechanics Models.
On the Quantum Density of States and Partitioning an Integer
This paper exploits the connection between the quantum many-particle density
of states and the partitioning of an integer in number theory. For bosons
in a one dimensional harmonic oscillator potential, it is well known that the
asymptotic (N -> infinity) density of states is identical to the
Hardy-Ramanujan formula for the partitions p(n), of a number n into a sum of
integers. We show that the same statistical mechanics technique for the density
of states of bosons in a power-law spectrum yields the partitioning formula for
p^s(n), the latter being the number of partitions of n into a sum of s-th
powers of a set of integers. By making an appropriate modification of the
statistical technique, we are also able to obtain d^s(n) for distinct
partitions. We find that the distinct square partitions d^2(n) show pronounced
oscillations as a function of n about the smooth curve derived by us. The
origin of these oscillations from the quantum point of view is discussed. After
deriving the Erdos-Lehner formula for restricted partitions for the case
by our method, we generalize it to obtain a new formula for distinct restricted
partitions.Comment: 17 pages including figure captions. 6 figures. To be submitted to J.
Phys. A: Math. Ge
Estimating the asymptotics of solid partitions
We study the asymptotic behavior of solid partitions using transition matrix
Monte Carlo simulations. If denotes the number of solid partitions of
an integer , we show that . This shows clear deviation from the value ,
attained by MacMahon numbers , that was conjectured to hold for solid
partitions as well. In addition, we find estimates for other sub-leading terms
in . In a pattern deviating from the asymptotics of line and plane
partitions, we need to add an oscillatory term in addition to the obvious
sub-leading terms. The period of the oscillatory term is proportional to
, the natural scale in the problem. This new oscillatory term might
shed some insight into why partitions in dimensions greater than two do not
admit a simple generating function.Comment: 21 pages, 8 figure
Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer
The number of solid partitions of a positive integer is an unsolved problem
in combinatorial number theory. In this paper, solid partitions are studied
numerically by the method of exact enumeration for integers up to 50 and by
Monte Carlo simulations using Wang-Landau sampling method for integers up to
8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where
p(n) is the number of solid partitions of the integer n. This result strongly
suggests that the MacMahon conjecture for solid partitions, though not exact,
could still give the correct leading asymptotic behaviour.Comment: 6 pages, 4 figures, revtex
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