2,111 research outputs found
Non-Euclidean geometry in nature
I describe the manifestation of the non-Euclidean geometry in the behavior of
collective observables of some complex physical systems. Specifically, I
consider the formation of equilibrium shapes of plants and statistics of sparse
random graphs. For these systems I discuss the following interlinked questions:
(i) the optimal embedding of plants leaves in the three-dimensional space, (ii)
the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to
chaotic Hamiltonian systems is adde
Statistics of reduced words in locally free and braid groups: Abstract studies and application to ballistic growth model
We study numerically and analytically the average length of reduced
(primitive) words in so-called locally free and braid groups. We consider the
situations when the letters in the initial words are drawn either without or
with correlations. In the latter case we show that the average length of the
reduced word can be increased or lowered depending on the type of correlation.
The ideas developed are used for analytical computation of the average number
of peaks of the surface appearing in some specific ballistic growth modelComment: 29 pages, LaTeX, 7 separated Postscript figures (available on
request), submitted to J. Phys. (A): Math. Ge
Correlation functions for some conformal theories on Riemann surfaces
We discuss the geometrical connection between 2D conformal field theories,
random walks on hyperbolic Riemann surfaces and knot theory. For the wide class
of CFTs with monodromies being the discrete subgroups of SL(2,R), the
determination of four-point correlation functions are related to construction
of topological invariants for random walks on multipunctured Riemann surfacesComment: 11 pages, LaTeX, 1 Postscript figur
Multifractality of entangled random walks and non-uniform hyperbolic spaces
Multifractal properties of the distribution of topological invariants for a
model of trajectories randomly entangled with a nonsymmetric lattice of
obstacles are investigated. Using the equivalence of the model to random walks
on a locally nonsymmetric tree, statistical properties of topological
invariants, such as drift and return probabilities, have been studied by means
of a renormalization group (RG) technique. The comparison of the analytical
RG--results with numerical simulations as well as with the rigorous results of
P.Gerl and W.Woess demonstrates clearly the validity of our approach. It is
shown explicitly by direct counting for the discrete version of the model and
by conformal methods for the continuous version that multifractality occurs
when local uniformity of the phase space (which has an exponentially large
number of states) has been broken.Comment: 28 pages, 11 eps-figures (enclosed
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