11,594 research outputs found
Large deviation asymptotics for occupancy problems
In the standard formulation of the occupancy problem one considers the
distribution of r balls in n cells, with each ball assigned independently to a
given cell with probability 1/n. Although closed form expressions can be given
for the distribution of various interesting quantities (such as the fraction of
cells that contain a given number of balls), these expressions are often of
limited practical use. Approximations provide an attractive alternative, and in
the present paper we consider a large deviation approximation as r and n tend
to infinity. In order to analyze the problem we first consider a dynamical
model, where the balls are placed in the cells sequentially and ``time''
corresponds to the number of balls that have already been thrown. A complete
large deviation analysis of this ``process level'' problem is carried out, and
the rate function for the original problem is then obtained via the contraction
principle. The variational problem that characterizes this rate function is
analyzed, and a fairly complete and explicit solution is obtained. The
minimizing trajectories and minimal cost are identified up to two constants,
and the constants are characterized as the unique solution to an elementary
fixed point problem. These results are then used to solve a number of
interesting problems, including an overflow problem and the partial coupon
collector's problem.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000013
Problems in extremal graph theory
We consider a variety of problems in extremal graph and set theory.
The {\em chromatic number} of , , is the smallest integer
such that is -colorable.
The {\it square} of , written , is the supergraph of in which also
vertices within distance 2 of each other in are adjacent.
A graph is a {\it minor} of if
can be obtained from a subgraph of by contracting edges.
We show that the upper bound for
conjectured by Wegner (1977) for planar graphs
holds when is a -minor-free graph.
We also show that is equal to the bound
only when contains a complete graph of that order.
One of the central problems of extremal hypergraph theory is
finding the maximum number of edges in a hypergraph
that does not contain a specific forbidden structure.
We consider as a forbidden structure a fixed number of members
that have empty common intersection
as well as small union.
We obtain a sharp upper bound on the size of uniform hypergraphs
that do not contain this structure,
when the number of vertices is sufficiently large.
Our result is strong enough to imply the same sharp upper bound
for several other interesting forbidden structures
such as the so-called strong simplices and clusters.
The {\em -dimensional hypercube}, ,
is the graph whose vertex set is and
whose edge set consists of the vertex pairs
differing in exactly one coordinate.
The generalized Tur\'an problem asks for the maximum number
of edges in a subgraph of a graph that does not contain
a forbidden subgraph .
We consider the Tur\'an problem where is and
is a cycle of length with .
Confirming a conjecture of Erd{\H o}s (1984),
we show that the ratio of the size of such a subgraph of
over the number of edges of is ,
i.e. in the limit this ratio approaches 0
as approaches infinity
Complexity of Preimage Problems for Deterministic Finite Automata
Given a subset of states S of a deterministic finite automaton and a word w, the preimage is the subset of all states that are mapped to a state from S by the action of w. We study the computational complexity of three problems related to the existence of words yielding certain preimages, which are especially motivated by the theory of synchronizing automata. The first problem is whether, for a given subset, there exists a word extending the subset (giving a larger preimage). The second problem is whether there exists a word totally extending the subset (giving the whole set of states) - it is equivalent to the problem whether there exists an avoiding word for the complementary subset. The third problem is whether there exists a word resizing the subset (giving a preimage of a different size). We also consider the variants of the problem where an upper bound on the length of the word is given in the input. Because in most cases our problems are computationally hard, we additionally consider parametrized complexity by the size of the given subset. We focus on the most interesting cases that are the subclasses of strongly connected, synchronizing, and binary automata
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
Enhanced Pairing of Quantum Critical Metals Near d=3+1
We study the dynamics of a quantum critical boson coupled to a Fermi surface
in intermediate energy regimes where the Landau damping of the boson can be
parametrically controlled, either via large Fermi velocity or by large N
techniques. We focus on developing a systematic approach to studying the BCS
instability, including careful treatment of the enhanced log^2 and log^3
singularities which appear already at 1-loop. We also treat possible
instabilities to charge density wave (CDW) formation, and compare the scales
Lambda_{BCS} and Lambda_{CDW} of the onset of the instabilities in different
parametric regimes. We address the question of whether the dressing of the
fermions into a non-Fermi liquid via interactions with the order parameter
field can happen at energies > Lambda_{BCS}, Lambda_{CDW}.Comment: 34 pages, 4 figures, submitted to PR
Composite fermi liquids in the lowest Landau level
We study composite fermi liquid (CFL) states in the lowest Landau level (LLL)
limit at a generic filling . We begin with the old
observation that, in compressible states, the composite fermion in the lowest
Landau level should be viewed as a charge-neutral particle carrying vorticity.
This leads to the absence of a Chern-Simons term in the effective theory of the
CFL. We argue here that instead a Berry curvature should be enclosed by the
fermi surface of composite fermions, with the total Berry phase fixed by the
filling fraction . We illustrate this point with the CFL of
fermions at filling fractions and (single and two-component) bosons
at . The Berry phase leads to sharp consequences in the transport
properties including thermal and spin Hall conductances, which in the RPA
approximation are distinct from the standard Halperin-Lee-Read predictions. We
emphasize that these results only rely on the LLL limit, and do not require
particle-hole symmetry, which is present microscopically only for fermions at
. Nevertheless, we show that the existing LLL theory of the composite
fermi liquid for bosons at does have an emergent particle-hole
symmetry. We interpret this particle-hole symmetry as a transformation between
the empty state at and the boson integer quantum hall state at .
This understanding enables us to define particle-hole conjugates of various
bosonic quantum Hall states which we illustrate with the bosonic Jain and
Pfaffian states. The bosonic particle-hole symmetry can be realized exactly on
the surface of a three-dimensional boson topological insulator. We also show
that with the particle-hole and spin rotation symmetries, there is no
gapped topological phase for bosons at .Comment: 16 pages, 1 figure, new version with minor change
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