54,352 research outputs found
A matrix perturbation view of the small world phenomenon
We use techniques from applied matrix analysis to study small world cutoff in a Markov chain. Our model consists of a periodic random walk plus uniform jumps. This has a direct interpretation as a teleporting random walk, of the type used by search engines to locate web pages, on a simple ring network. More loosely, the model may be regarded as an analogue of the original small world network of Watts and Strogatz [Nature, 393 (1998), pp. 440-442]. We measure the small world property by expressing the mean hitting time, averaged over all states, in terms of the expected number of shortcuts per random walk. This average mean hitting time is equivalent to the expected number of steps between a pair of states chosen uniformly at random. The analysis involves nonstandard matrix perturbation theory and the results come with rigorous and sharp asymptotic error estimates. Although developed in a different context, the resulting cutoff diagram agrees closely with that arising from the mean-field network theory of Newman, Moore, and Watts [Phys. Rev. Lett., 84 (2000), pp. 3201-3204]
A Note on Background (In)dependence
In general quantum systems there are two kinds of spacetime modes, those that
fluctuate and those that do not. Fluctuating modes have normalizable
wavefunctions. In the context of 2D gravity and ``non-critical'' string theory
these are called macroscopic states. The theory is independent of the initial
Euclidean background values of these modes. Non-fluctuating modes have
non-normalizable wavefunctions and correspond to microscopic states. The theory
depends on the background value of these non-fluctuating modes, at least to all
orders in perturbation theory. They are superselection parameters and should
not be minimized over. Such superselection parameters are well known in field
theory. Examples in string theory include the couplings (including the
cosmological constant) in the matrix models and the mass of the two-dimensional
Euclidean black hole. We use our analysis to argue for the finiteness of the
string perturbation expansion around these backgrounds.Comment: 16 page
The Self-Trapping Line of the Holstein Molecular Crystal Model in One Dimension
The ground state of the Holstein molecular crystal model in one dimension is
studied using the Global-Local variational method, analyzing in particular the
total energy, kinetic energy, phonon energy, and interaction energy over a
broad region of the polaron parameter space. Through the application of
objective criteria, a unique curve is identified that simply, accurately, and
robustly locates the self-trapping transition separating small polaron and
large polaron behavior
Heavy Quark Effective Theory beyond Perturbation Theory: Renormalons, the Pole Mass and the Residual Mass Term
We study the asymptotic behaviour of the perturbative series in the heavy
quark effective theory (HQET) using the expansion. We find that this
theory suffers from an {\it ultraviolet} renormalon problem, corresponding to a
non-Borel-summable behaviour of perturbation series in large orders, and
leading to a principal nonperturbative ambiguity in its definition. This
ambiguity is related to an {\it infrared} renormalon in the pole mass and can
be understood as the necessity to include the residual mass term in
the definition of HQET, which must be considered as ambiguous (and possibly
complex), and is required to cancel the ultraviolet renormalon singularity
generated by the perturbative expansion. The formal status of is
thus identical to that of condensates in the conventional short-distance
expansion of correlation functions in QCD. The status of the pole mass of a
heavy quark, the operator product expansion for inclusive decays, and QCD sum
rules in the HQET are discussed in this context.Comment: LATEX, 43 pages, 6 figures appended as uu-encoded file, MPI-PhT/94-9,
(Text as to appear in NPB, typing errors corrected [Eq.(3.24),(3.26)], some
statements in Sect.5 more precise
On the foundation of equilibrium quantum statistical mechanics
We discuss the condition for the validity of equilibrium quantum statistical
mechanics in the light of recent developments in the understanding of classical
and quantum chaotic motion. In particular, the ergodicity parameter is shown to
provide the conditions under which quantum statistical distributions can be
derived from the quantum dynamics of a classical ergodic Hamiltonian system.Comment: 10 pages (RevTeX), 2 eps figure
Non-Perturbative Effects in 2-D String Theory or Beyond the Liouville Wall
We discuss continuous and discrete sectors in the collective field theory of
matrix models. A canonical Lorentz invariant field theory extension of
collective field theory is presented and its classical solutions in Euclidean
and Minkowski space are found. We show that the discrete, low density, sector
of collective field theory includes single eigenvalue Euclidean instantons
which tunnel between different vacua of the extended theory. We further show
that these ``stringy" instantons induce non-perturbative effective operators of
strength in the extended theory. The relationship of the
world sheet description of string theory and Liouville theory to the effective
space-time theory is explained. We also comment on the role of the discrete,
low density, sector of collective field theory in that framework.Comment: 44 pages, 9 figures available as eps files on reques
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