88,072 research outputs found
Random non-crossing plane configurations: A conditioned Galton-Watson tree approach
We study various models of random non-crossing configurations consisting of
diagonals of convex polygons, and focus in particular on uniform dissections
and non-crossing trees. For both these models, we prove convergence in
distribution towards Aldous' Brownian triangulation of the disk. In the case of
dissections, we also refine the study of the maximal vertex degree and validate
a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of
an underlying Galton-Watson tree structure.Comment: 24 pages, 9 figure
Fixed parameter tractability of crossing minimization of almost-trees
We investigate exact crossing minimization for graphs that differ from trees
by a small number of additional edges, for several variants of the crossing
minimization problem. In particular, we provide fixed parameter tractable
algorithms for the 1-page book crossing number, the 2-page book crossing
number, and the minimum number of crossed edges in 1-page and 2-page book
drawings.Comment: Graph Drawing 201
Wall-Crossing from Boltzmann Black Hole Halos
A key question in the study of N=2 supersymmetric string or field theories is
to understand the decay of BPS bound states across walls of marginal stability
in the space of parameters or vacua. By representing the potentially unstable
bound states as multi-centered black hole solutions in N=2 supergravity, we
provide two fully general and explicit formulae for the change in the (refined)
index across the wall. The first, "Higgs branch" formula relies on Reineke's
results for invariants of quivers without oriented loops, specialized to the
Abelian case. The second, "Coulomb branch" formula results from evaluating the
symplectic volume of the classical phase space of multi-centered solutions by
localization. We provide extensive evidence that these new formulae agree with
each other and with the mathematical results of Kontsevich and Soibelman (KS)
and Joyce and Song (JS). The main physical insight behind our results is that
the Bose-Fermi statistics of individual black holes participating in the bound
state can be traded for Maxwell-Boltzmann statistics, provided the (integer)
index \Omega(\gamma) of the internal degrees of freedom carried by each black
hole is replaced by an effective (rational) index \bar\Omega(\gamma)=
\sum_{m|\gamma} \Omega(\gamma/m)/m^2. A similar map also exists for the refined
index. This observation provides a physical rationale for the appearance of the
rational Donaldson-Thomas invariant \bar\Omega(\gamma) in the works of KS and
JS. The simplicity of the wall crossing formula for rational invariants allows
us to generalize the "semi-primitive wall-crossing formula" to arbitrary decays
of the type \gamma\to M\gamma_1+N\gamma_2 with M=2,3.Comment: 71 pages, 1 figure; v3: changed normalisation of symplectic form
3.22, corrected 3.35, other cosmetic change
Production matrices for geometric graphs
We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot numbers, and Fibonacci numbers arise in a natural way, and also new formulas are obtained, such as a formula for the number of non-crossing geometric graphs with root vertex of given degree. The characteristic polynomials of some of these production matrices are also presented. The proofs make use of generating trees and Riordan arrays.Postprint (updated version
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