2,940 research outputs found
Rearranged Stochastic Heat Equation
The purpose of this work is to provide an explicit construction of a strong
Feller semigroup on the space of probability measures over the real line that
additionally maps bounded measurable functions into Lipschitz continuous
functions, with a Lipschitz constant that blows up in an integrable manner in
small time. Our construction relies on a rearranged version of the stochastic
heat equation on the circle driven by a coloured noise. Formally, this
stochastic equation writes as a reflected equation in infinite dimension, a
topic that is known to be challenging. Under the action of the rearrangement,
the solution is forced to live in a space of quantile functions that is
isometric to the space of probability measures on the real line. We prove the
equation to be solvable by means of an Euler scheme in which we alternate flat
dynamics in the space of random variables on the circle with a rearrangement
operation that projects back the random variables onto the subset of quantile
functions. A first challenge is to prove that this scheme is tight. A second
one is to provide a consistent theory for the limiting reflected equation and
in particular to interpret in a relevant manner the reflection term. The last
step in our work is to establish the aforementioned Lipschitz property of the
semigroup by adapting earlier ideas from the Bismut-Elworthy-Li formula in
stochastic analysis
Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries
Using exact computations we study the classical hard-core monomer-dimer
models on m x n plane lattice strips with free boundaries. For an arbitrary
number v of monomers (or vacancies), we found a logarithmic correction term in
the finite-size correction of the free energy. The coefficient of the
logarithmic correction term depends on the number of monomers present (v) and
the parity of the width n of the lattice strip: the coefficient equals to v
when n is odd, and v/2 when n is even. The results are generalizations of the
previous results for a single monomer in an otherwise fully packed lattice of
dimers.Comment: 4 pages, 2 figure
Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density
The classical monomer-dimer model in two-dimensional lattices has been shown
to belong to the \emph{``#P-complete''} class, which indicates the problem is
computationally ``intractable''. We use exact computational method to
investigate the number of ways to arrange dimers on
two-dimensional rectangular lattice strips with fixed dimer density . For
any dimer density , we find a logarithmic correction term in the
finite-size correction of the free energy per lattice site. The coefficient of
the logarithmic correction term is exactly -1/2. This logarithmic correction
term is explained by the newly developed asymptotic theory of Pemantle and
Wilson. The sequence of the free energy of lattice strips with cylinder
boundary condition converges so fast that very accurate free energy
for large lattices can be obtained. For example, for a half-filled lattice,
, while and . For , is accurate at least to 10 decimal
digits. The function reaches the maximum value at , with 11 correct digits. This is also
the \md constant for two-dimensional rectangular lattices. The asymptotic
expressions of free energy near close packing are investigated for finite and
infinite lattice widths. For lattices with finite width, dependence on the
parity of the lattice width is found. For infinite lattices, the data support
the functional form obtained previously through series expansions.Comment: 15 pages, 5 figures, 5 table
Dual Monte Carlo and Cluster Algorithms
We discuss the development of cluster algorithms from the viewpoint of
probability theory and not from the usual viewpoint of a particular model. By
using the perspective of probability theory, we detail the nature of a cluster
algorithm, make explicit the assumptions embodied in all clusters of which we
are aware, and define the construction of free cluster algorithms. We also
illustrate these procedures by rederiving the Swendsen-Wang algorithm,
presenting the details of the loop algorithm for a worldline simulation of a
quantum 1/2 model, and proposing a free cluster version of the
Swendsen-Wang replica method for the random Ising model. How the principle of
maximum entropy might be used to aid the construction of cluster algorithms is
also discussed.Comment: 25 pages, 4 figures, to appear in Phys.Rev.
Weighted distances in scale-free preferential attachment models
We study three preferential attachment models where the parameters are such
that the asymptotic degree distribution has infinite variance. Every edge is
equipped with a non-negative i.i.d. weight. We study the weighted distance
between two vertices chosen uniformly at random, the typical weighted distance,
and the number of edges on this path, the typical hopcount. We prove that there
are precisely two universality classes of weight distributions, called the
explosive and conservative class. In the explosive class, we show that the
typical weighted distance converges in distribution to the sum of two i.i.d.
finite random variables. In the conservative class, we prove that the typical
weighted distance tends to infinity, and we give an explicit expression for the
main growth term, as well as for the hopcount. Under a mild assumption on the
weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear
in Random Structures and Algorithm
Enumeration of self avoiding trails on a square lattice using a transfer matrix technique
We describe a new algebraic technique, utilising transfer matrices, for
enumerating self-avoiding lattice trails on the square lattice. We have
enumerated trails to 31 steps, and find increased evidence that trails are in
the self-avoiding walk universality class. Assuming that trails behave like , we find and .Comment: To be published in J. Phys. A:Math Gen. Pages: 16 Format: RevTe
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Enantioselective PCCP Brønsted acid-catalyzed aza-Piancatelli rearrangement.
An enantioselective aza-Piancatelli rearrangement has been developed using a chiral Brønsted acid based on pentacarboxycyclopentadiene (PCCP). This reaction provides rapid access to valuable chiral 4-amino-2-cyclopentenone building blocks from readily available starting material and is operationally simple
New Lower Bounds on the Self-Avoiding-Walk Connective Constant
We give an elementary new method for obtaining rigorous lower bounds on the
connective constant for self-avoiding walks on the hypercubic lattice .
The method is based on loop erasure and restoration, and does not require exact
enumeration data. Our bounds are best for high , and in fact agree with the
first four terms of the expansion for the connective constant. The bounds
are the best to date for dimensions , but do not produce good results
in two dimensions. For , respectively, our lower bound is within
2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.Comment: 35 pages, 388480 bytes Postscript, NYU-TH-93/02/0
A major star formation region in the receding tip of the stellar Galactic bar
We present an analysis of the optical spectroscopy of 58 stars in the
Galactic plane at \arcdeg, where a prominent excess in the flux
distribution and star counts have been observed in several spectral regions, in
particular in the Two Micron Galactic Survey (TMGS) catalog. The sources were
selected from the TMGS, to have a magnitude brighter than +5 mag and be
within 2 degrees of the Galactic plane. More than 60% of the spectra correspond
to stars of luminosity class I, and a significant proportion of the remainder
are very late giants which would also be fast evolving. This very high
concentration of young sources points to the existence of a major star
formation region in the Galactic plane, located just inside the assumed origin
of the Scutum spiral arm. Such regions can form due to the concentrations of
shocked gas where a galactic bar meets a spiral arm, as is observed at the ends
of the bars of face-on external galaxies. Thus, the presence of a massive star
formation region is very strong supporting evidence for the presence of a bar
in our Galaxy.Comment: 13 pages (latex) + 4 figures (eps), accepted in ApJ Let
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