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Rank Bounded Hibi Subrings for Planar Distributive Lattices
Let be a distributive lattice and the associated Hibi ring. We
show that if is planar, then any bounded Hibi subring of has a
quadratic Gr\"obner basis. We characterize all planar distributive lattices
for which any proper rank bounded Hibi subring of has a linear
resolution. Moreover, if is linearly related for a lattice , we find
all the rank bounded Hibi subrings of which are linearly related too.Comment: Accepted in Mathematical Communication
Probability Distribution of the Shortest Path on the Percolation Cluster, its Backbone and Skeleton
We consider the mean distribution functions Phi(r|l), Phi(B)(r|l), and
Phi(S)(r|l), giving the probability that two sites on the incipient percolation
cluster, on its backbone and on its skeleton, respectively, connected by a
shortest path of length l are separated by an Euclidean distance r. Following a
scaling argument due to de Gennes for self-avoiding walks, we derive analytical
expressions for the exponents g1=df+dmin-d and g1B=g1S-3dmin-d, which determine
the scaling behavior of the distribution functions in the limit x=r/l^(nu) much
less than 1, i.e., Phi(r|l) proportional to l^(-(nu)d)x^(g1), Phi(B)(r|l)
proportional to l^(-(nu)d)x^(g1B), and Phi(S)(r|l) proportional to
l^(-(nu)d)x^(g1S), with nu=1/dmin, where df and dmin are the fractal dimensions
of the percolation cluster and the shortest path, respectively. The theoretical
predictions for g1, g1B, and g1S are in very good agreement with our numerical
results.Comment: 10 pages, 3 figure
Correlations in Ising chains with non-integrable interactions
Two-spin correlations generated by interactions which decay with distance r
as r^{-1-sigma} with -1 <sigma <0 are calculated for periodic Ising chains of
length L. Mean-field theory indicates that the correlations, C(r,L), diminish
in the thermodynamic limit L -> \infty, but they contain a singular structure
for r/L -> 0 which can be observed by introducing magnified correlations,
LC(r,L)-\sum_r C(r,L). The magnified correlations are shown to have a scaling
form F(r/L) and the singular structure of F(x) for x->0 is found to be the same
at all temperatures including the critical point. These conclusions are
supported by the results of Monte Carlo simulations for systems with sigma
=-0.50 and -0.25 both at the critical temperature T=Tc and at T=2Tc.Comment: 13 pages, latex, 5 eps figures in a separate uuencoded file, to
appear in Phys.Rev.
Regularity of joint-meet ideals of distributive lattices
Let be a distributive lattice and the associated Hibi ring. We
compute \reg R(L) when is a planar lattice and give a lower bound for
\reg R(L) when is non-planar, in terms of the combinatorial data of
As a consequence, we characterize the distributive lattices for which the
associated Hibi ring has a linear resolution
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