45,148 research outputs found
Asymptotics in randomized urn models
This paper studies a very general urn model stimulated by designs in clinical
trials, where the number of balls of different types added to the urn at trial
n depends on a random outcome directed by the composition at trials
1,2,...,n-1. Patient treatments are allocated according to types of balls. We
establish the strong consistency and asymptotic normality for both the urn
composition and the patient allocation under general assumptions on random
generating matrices which determine how balls are added to the urn. Also we
obtain explicit forms of the asymptotic variance-covariance matrices of both
the urn composition and the patient allocation. The conditions on the
nonhomogeneity of generating matrices are mild and widely satisfied in
applications. Several applications are also discussed.Comment: Published at http://dx.doi.org/10.1214/105051604000000774 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Majorana Fermions, Exact Mappings between Classical and Topological Orders
Motivated by the duality between site-centered spin and bond-centered spin in
one-dimensional system, which connects two different constructions of fermions
from the same set of Majorana fermions, we show that two-dimensional models
with topological orders can be constructed from certain well-known models with
classical orders characterized by symmetry-breaking. Topology-dependent ground
state degeneracy, vanishing two-point correlation functions, and unpaired
Majorana fermions on boundaries emerge naturally from such construction. The
approach opens a new way to construct and characterize topological orders.Comment: 5 pages, 4 figure
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