1,079 research outputs found
Making Markov chains less lazy
The mixing time of an ergodic, reversible Markov chain can be bounded in
terms of the eigenvalues of the chain: specifically, the second-largest
eigenvalue and the smallest eigenvalue. It has become standard to focus only on
the second-largest eigenvalue, by making the Markov chain "lazy". (A lazy chain
does nothing at each step with probability at least 1/2, and has only
nonnegative eigenvalues.)
An alternative approach to bounding the smallest eigenvalue was given by
Diaconis and Stroock and Diaconis and Saloff-Coste. We give examples to show
that using this approach it can be quite easy to obtain a bound on the smallest
eigenvalue of a combinatorial Markov chain which is several orders of magnitude
below the best-known bound on the second-largest eigenvalue.Comment: 8 page
Toric algebra of hypergraphs
The edges of any hypergraph parametrize a monomial algebra called the edge
subring of the hypergraph. We study presentation ideals of these edge subrings,
and describe their generators in terms of balanced walks on hypergraphs. Our
results generalize those for the defining ideals of edge subrings of graphs,
which are well-known in the commutative algebra community, and popular in the
algebraic statistics community. One of the motivations for studying toric
ideals of hypergraphs comes from algebraic statistics, where generators of the
toric ideal give a basis for random walks on fibers of the statistical model
specified by the hypergraph. Further, understanding the structure of the
generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in
algebraic statistics and to combinatorial discrepancy. Section 6 (open
problems) has been moderately revise
Adiabatic Quantum State Generation and Statistical Zero Knowledge
The design of new quantum algorithms has proven to be an extremely difficult
task. This paper considers a different approach to the problem, by studying the
problem of 'quantum state generation'. This approach provides intriguing links
between many different areas: quantum computation, adiabatic evolution,
analysis of spectral gaps and groundstates of Hamiltonians, rapidly mixing
Markov chains, the complexity class statistical zero knowledge, quantum random
walks, and more.
We first show that many natural candidates for quantum algorithms can be cast
as a state generation problem. We define a paradigm for state generation,
called 'adiabatic state generation' and develop tools for adiabatic state
generation which include methods for implementing very general Hamiltonians and
ways to guarantee non negligible spectral gaps. We use our tools to prove that
adiabatic state generation is equivalent to state generation in the standard
quantum computing model, and finally we show how to apply our techniques to
generate interesting superpositions related to Markov chains.Comment: 35 pages, two figure
Fractal dimension of domain walls in the Edwards-Anderson spin glass model
We study directly the length of the domain walls (DW) obtained by comparing
the ground states of the Edwards-Anderson spin glass model subject to periodic
and antiperiodic boundary conditions. For the bimodal and Gaussian bond
distributions, we have isolated the DW and have calculated directly its fractal
dimension . Our results show that, even though in three dimensions
is the same for both distributions of bonds, this is clearly not the case for
two-dimensional (2D) systems. In addition, contrary to what happens in the case
of the 2D Edwards-Anderson spin glass with Gaussian distribution of bonds, we
find no evidence that the DW for the bimodal distribution of bonds can be
described as a Schramm-Loewner evolution processes.Comment: 6 pages, 5 figures. Accepted for publication in PR
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