42 research outputs found
Fully Analyzing an Algebraic Polya Urn Model
This paper introduces and analyzes a particular class of Polya urns: balls
are of two colors, can only be added (the urns are said to be additive) and at
every step the same constant number of balls is added, thus only the color
compositions varies (the urns are said to be balanced). These properties make
this class of urns ideally suited for analysis from an "analytic combinatorics"
point-of-view, following in the footsteps of Flajolet-Dumas-Puyhaubert, 2006.
Through an algebraic generating function to which we apply a multiple
coalescing saddle-point method, we are able to give precise asymptotic results
for the probability distribution of the composition of the urn, as well as
local limit law and large deviation bounds.Comment: LATIN 2012, Arequipa : Peru (2012
Dynamic Many-Body Theory. II. Dynamics of Strongly Correlated Fermi Fluids
We develop a systematic theory of multi-particle excitations in strongly
interacting Fermi systems. Our work is the generalization of the time-honored
work by Jackson, Feenberg, and Campbell for bosons, that provides, in its most
advanced implementation, quantitative predictions for the dynamic structure
function in the whole experimentally accessible energy/momentum regime. Our
view is that the same physical effects -- namely fluctuations of the wave
function at an atomic length scale -- are responsible for the correct
energetics of the excitations in both Bose and Fermi fluids. Besides a
comprehensive derivation of the fermion version of the theory and discussion of
the approximations made, we present results for homogeneous He-3 and electrons
in three dimensions. We find indeed a significant lowering of the zero sound
mode in He-3 and a broadening of the collective mode due to the coupling to
particle-hole excitations in good agreement with experiments. The most visible
effect in electronic systems is the appearance of a ``double-plasmon''
excitation.Comment: submitted to Phys. Rev.
Cutting edges at random in large recursive trees
We comment on old and new results related to the destruction of a random
recursive tree (RRT), in which its edges are cut one after the other in a
uniform random order. In particular, we study the number of steps needed to
isolate or disconnect certain distinguished vertices when the size of the tree
tends to infinity. New probabilistic explanations are given in terms of the
so-called cut-tree and the tree of component sizes, which both encode different
aspects of the destruction process. Finally, we establish the connection to
Bernoulli bond percolation on large RRT's and present recent results on the
cluster sizes in the supercritical regime.Comment: 29 pages, 3 figure
Kirkman's Hypothesis Revisited
Watson proved Kirkman’s hypothesis (partially solved by Cayley). Using Lagrange Inversion, we drastically shorten Watson’s computations and generalize his results at the same time
Asymptotic results for the number of paths in a grid
In two recent papers, Albrecht and White ['Counting paths in a grid', Austral. Math. Soc. Gaz. 35 (2008), 43-48] and Hirschhorn ['Comment on "Counting paths in a grid", Austral. Math. Soc. Gaz. 36 (2009), 50-52] considered the problem of counting the total number Pm,n of certain restricted lattice paths in an m × n grid of cells, which appeared in the context of counting train paths through a rail network. Here we give a precise study of the asymptotic behaviour of these numbers for the square grid, extending the results of Hirschhorn, and furthermore provide an asymptotic equivalent of these numbers for a rectangular grid with a constant proportion α = m/n between the side lengths. © 2011 Australian Mathematical Publishing Association Inc
A short proof of a series evaluation in terms of harmonic numbers
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