96 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
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
From statistics of regular tree-like graphs to distribution function and gyration radius of branched polymers
We consider flexible branched polymer, with quenched branch structure, and
show that its conformational entropy as a function of its gyration radius ,
at large , obeys, in the scaling sense, , with
bond length (or Kuhn segment) and defined as an average spanning distance.
We show that this estimate is valid up to at most the logarithmic correction
for any tree. We do so by explicitly computing the largest eigenvalues of
Kramers matrices for both regular and "sparse" 3-branched trees, uncovering on
the way their peculiar mathematical properties.Comment: 9 pages, 4 figure
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