The critical random graph,with martingales

We give a short proof that the largest component C ₁ of the random graph G(n, 1/n) is of size approximately n ^2/3. The proof gives explicit bounds for the probability that the ratio is very large or very small. In particular, the probability that n ^−2/3|C ₁| exceeds A is at most e ^{- cA3}{e^{ - c{A^3}}} for some c > 0.     

Scaling limits for internal aggregation models with multiple sources

We study the scaling limits of three different aggregation models on ℤ ^d : internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in ℝ ^d . In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.     

Electron-pair resonance in the coulomb blockade

We study many-body corrections to the cotunneling current via a localized state with energy epsilon(d) at large bias voltages V. We show that the transfer of electron pairs, enabled by the Coulomb repulsion in the localized level, results in ionization resonance peaks in the third derivative of the current with respect to V, centered at eV=+/-2epsilon(d)/3. Our results predict the existence of previously unnoticed structure within Coulomb-blockade diamonds.     

Comments on "A New CT-Aided Stereotactic Neurosurgery Technique"

The method given for finding the position of a target in three-dimensional space from a CT image is incorrect. The solution is based on representation of the target vector as a linear combination of two coplanar nonparallel vectors. This results in a single linear equation with two unknowns. An alternative formulation is offered, solved, and the existence and uniqueness of the solution is proved.     

The problems in quantum foundations in the light of gauge theories

We review the issues of nonseparability and seemingly acausal propagation of information in EPR, as displayed by experiments and the failure of Bell's inequalities. We show that global effects are in the very nature of the geometric structure of modern physical theories, occurring even at the classical level. The Aharonov-Bohm effect, magnetic monopoles, instantons, etc. result from the topology and homotopy features of the fiber bundle manifolds of gauge theories. The conservation of probabilities, a supposedly highly quantum effect, is also achieved through global geometry equations. The EPR observables all fit in such geometries, and space-time is a truncated representation and is not the correct arena for their understanding. Relativistic quantum field theory represents the global action of the measurement operators as the zero-momentum (and therefore spatially infinitely spread) limit of their wave functions (form factors). We also analyze the collapse of the state vector as a case of spontaneous symmetry breakdown in the apparatus-observed state interaction.Work supported in part by U.S. DOE Grant DE-FG05-85ER40200.Wolfson Chair Extraordinary in Theoretical Physics, TAUP N-161-85.     

On the symmetric square Unstable twisted characters

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads — see [FK] — to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in [FK] is the unstable one. A direct local proof of the fundamental lemma is given in [V].     

A simple derivation of Borland's locality result

A new proof of Borland's localization theorem is given, which is simpler, less rigorous, but gives more quantitative results.     

Gravity from symmetry breakdown of a gauge affine theory

We construct a gauge field theory based on GA(4, ) space-time symmetry containing spinorial and tensorial (infinite-component) matter manifields. We break the dilation symmetry and thus trigger a spontaneous breaking of the SL(4, )/SO(1, 3), also generating Newton's constant. The resulting ‘large-scale” space-time (a compared to Planck length) is of Riemann-Cartan type and in the flat limit we recover special relativity.     

Quantizing gravity and spacetime: Where do we stand ?

We review five possible solutions to the riddle posed by Quantum Gravity: (1) Gravity should stay as a classical theory (L. Rosenfeld); (2) Quantum Gravity requires a formalism which will take the human mind (or the intelligent observer) into account, resolving at the same time the riddle of the collapse of the wave function/state vector in Quantum Mechanics in general (Penrose); (3) Perturbative Quantization; (4) Hamiltonian Quantization (Dirac, Ashtekar); (5) String Theory. We also discuss the quantization of spacetime.