563 research outputs found
Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
We develop a scaling theory for KPZ growth in one dimension by a detailed
study of the polynuclear growth (PNG) model. In particular, we identify three
universal distributions for shape fluctuations and their dependence on the
macroscopic shape. These distribution functions are computed using the
partition function of Gaussian random matrices in a cosine potential.Comment: 4 pages, 3 figures, 1 table, RevTeX, revised version, accepted for
publication in PR
Condensation of the roots of real random polynomials on the real axis
We introduce a family of real random polynomials of degree n whose
coefficients a_k are symmetric independent Gaussian variables with variance
= e^{-k^\alpha}, indexed by a real \alpha \geq 0. We compute exactly
the mean number of real roots for large n. As \alpha is varied, one finds
three different phases. First, for 0 \leq \alpha \sim
(\frac{2}{\pi}) \log{n}. For 1 < \alpha < 2, there is an intermediate phase
where grows algebraically with a continuously varying exponent,
\sim \frac{2}{\pi} \sqrt{\frac{\alpha-1}{\alpha}} n^{\alpha/2}. And finally for
\alpha > 2, one finds a third phase where \sim n. This family of real
random polynomials thus exhibits a condensation of their roots on the real line
in the sense that, for large n, a finite fraction of their roots /n are
real. This condensation occurs via a localization of the real roots around the
values \pm \exp{[\frac{\alpha}{2}(k+{1/2})^{\alpha-1} ]}, 1 \ll k \leq n.Comment: 13 pages, 2 figure
Random tree growth by vertex splitting
We study a model of growing planar tree graphs where in each time step we
separate the tree into two components by splitting a vertex and then connect
the two pieces by inserting a new link between the daughter vertices. This
model generalises the preferential attachment model and Ford's -model
for phylogenetic trees. We develop a mean field theory for the vertex degree
distribution, prove that the mean field theory is exact in some special cases
and check that it agrees with numerical simulations in general. We calculate
various correlation functions and show that the intrinsic Hausdorff dimension
can vary from one to infinity, depending on the parameters of the model.Comment: 47 page
Heteroepitaxial growth of ferromagnetic MnSb(0001) films on Ge/Si(111) virtual substrates
Molecular beam epitaxial growth of ferromagnetic MnSb(0001) has been achieved on high quality, fully relaxed Ge(111)/Si(111) virtual substrates grown by reduced pressure chemical vapor deposition. The epilayers were characterized using reflection high energy electron diffraction, synchrotron hard X-ray diffraction, X-ray photoemission spectroscopy, and magnetometry. The surface reconstructions, magnetic properties, crystalline quality, and strain relaxation behavior of the MnSb films are similar to those of MnSb grown on GaAs(111). In contrast to GaAs substrates, segregation of substrate atoms through the MnSb film does not occur, and alternative polymorphs of MnSb are absent
Expected length of the longest common subsequence for large alphabets
We consider the length L of the longest common subsequence of two randomly
uniformly and independently chosen n character words over a k-ary alphabet.
Subadditivity arguments yield that the expected value of L, when normalized by
n, converges to a constant C_k. We prove a conjecture of Sankoff and Mainville
from the early 80's claiming that C_k\sqrt{k} goes to 2 as k goes to infinity.Comment: 14 pages, 1 figure, LaTe
An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution
We compute exactly the asymptotic distribution of scaled height in a
(1+1)--dimensional anisotropic ballistic deposition model by mapping it to the
Ulam problem of finding the longest nondecreasing subsequence in a random
sequence of integers. Using the known results for the Ulam problem, we show
that the scaled height in our model has the Tracy-Widom distribution appearing
in the theory of random matrices near the edges of the spectrum. Our result
supports the hypothesis that various growth models in dimensions that
belong to the Kardar-Parisi-Zhang universality class perhaps all share the same
universal Tracy-Widom distribution for the suitably scaled height variables.Comment: 5 pages Revtex, 3 .eps figures included, new references adde
Kang-Redner Anomaly in Cluster-Cluster Aggregation
The large time, small mass, asymptotic behavior of the average mass
distribution \pb is studied in a -dimensional system of diffusing
aggregating particles for . By means of both a renormalization
group computation as well as a direct re-summation of leading terms in the
small reaction-rate expansion of the average mass distribution, it is shown
that \pb \sim \frac{1}{t^d} (\frac{m^{1/d}}{\sqrt{t}})^{e_{KR}} for , where and . In two
dimensions, it is shown that \pb \sim \frac{\ln(m) \ln(t)}{t^2} for . Numerical simulations in two dimensions supporting the analytical
results are also presented.Comment: 11 pages, 6 figures, Revtex
Real roots of Random Polynomials: Universality close to accumulation points
We identify the scaling region of a width O(n^{-1}) in the vicinity of the
accumulation points of the real roots of a random Kac-like polynomial
of large degree n. We argue that the density of the real roots in this region
tends to a universal form shared by all polynomials with independent,
identically distributed coefficients c_i, as long as the second moment
\sigma=E(c_i^2) is finite. In particular, we reveal a gradual (in contrast to
the previously reported abrupt) and quite nontrivial suppression of the number
of real roots for coefficients with a nonzero mean value \mu_n = E(c_i) scaled
as \mu_n\sim n^{-1/2}.Comment: Some minor mistakes that crept through into publication have been
removed. 10 pages, 12 eps figures. This version contains all updates, clearer
pictures and some more thorough explanation
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