1,917 research outputs found
The Effect of Interaction on Shot Noise in The Quantum Limit
We employ a non-linear sigma model defined on a Keldysh contour to study the
current and the current noise in a diffusive micro-bridge in the presence of
electron-electron interactions. Out of equilibrium the fluctuation-dissipation
theorem (FDT) does not apply, hence these two quantities are not simply
interrelated. For a two-dimensional electron gas (2DEG) we obtain logarithmic
singularities in the low frequency limit. PACS Nos. 71.10.Ay, 71.23.An,
73.50.TdComment: Submitted to the proceedings of 36 Renconters de Moriond "Electronic
correlations: from meso- to nano-physics". One figure adde
Card shuffling and diophantine approximation
The ``overlapping-cycles shuffle'' mixes a deck of cards by moving either
the th card or the th card to the top of the deck, with probability
half each. We determine the spectral gap for the location of a single card,
which, as a function of and , has surprising behavior. For example,
suppose is the closest integer to for a fixed real
. Then for rational the spectral gap is
, while for poorly approximable irrational numbers ,
such as the reciprocal of the golden ratio, the spectral gap is
.Comment: Published in at http://dx.doi.org/10.1214/07-AAP484 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Tug-of-war and the infinity Laplacian
We prove that every bounded Lipschitz function F on a subset Y of a length
space X admits a tautest extension to X, i.e., a unique Lipschitz extension u
for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do
not intersect Y.
This was previously known only for bounded domains R^n, in which case u is
infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also
prove the first general uniqueness results for Delta_infty u = g on bounded
subsets of R^n (when g is uniformly continuous and bounded away from zero), and
analogous results for bounded length spaces.
The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be
the value of the following two-player zero-sum game, called tug-of-war: fix
x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner
chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y,
and player one's payoff is
F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i)
We show that the u^\epsilon converge uniformly to u as epsilon tends to zero.
Even for bounded domains in R^n, the game theoretic description of
infinity-harmonic functions yields new intuition and estimates; for instance,
we prove power law bounds for infinity-harmonic functions in the unit disk with
boundary values supported in a delta-neighborhood of a Cantor set on the unit
circle.Comment: 44 pages, 4 figure
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