Triangle percolation in mean field random graphs -- with PDE

We apply a PDE-based method to deduce the critical time and the size of the giant component of the ``triangle percolation'' on the Erd\H{o}s-R\'enyi random graph process investigated by Palla, Der\'enyi and VicsekComment: Summary of the changes made: We have changed a remark about k-clique percolation in the first paragraph. Two new paragraphs are inserted after equation (4.4) with two applications of the equation. We have changed the names of some variables in our formula

Exponential Upper Bounds via Martingales for Multiplexers with Markovian Arrivals.

We obtain explicit upper bounds in closed form for the queue length in a slotted time FCFS queue in which the service requirement is a sum of independent Markov processes on the state space {O, 1}, with integral service rate. The bound is of the form P[queue length ≥ b] ≤ cy^(-b) for any b ≥ 1 where c 1 are given explicitly in terms of the parameters of the model. The model can be viewed as an approximation for the burst-level component of the queue in an ATM multiplexer. We obtain heavy traffic bounds for the mean queue length and show that for typical parameters this far exceeds the mean queue length for independent arrivals at the same load. We compare our results on the mean queue length with an analytic expression for the case of unit service rate, and compare our results on the full distribution with computer simulations

A Model of Continuous Polymers with Random Charges

We study a model of polymers with random charges; the possible shapes of the polymer are represented by the sample paths of a Brownian motion, and the cumulative charge distribution along a polymer is modelled by a realisation of a Brownian bridge. Charges interact through a general positive-definite two-body potential. We study the infinite volume free energy density for a fixed realisation of the Brownian motion; this is not self-averaging, but shows on the contrary a sample dependence through the local time of the Brownian motion. We obtain an explicit series representation for the free energy density; this has a finite radius of convergence, but the free energy is nevertheless analytic in the inverse temperature in the physical domain

Proof of Bose-Einstein Condensation for Interacting Gases with a One-Particle Spectral Gap

Using a specially tuned mean-field Bose gas as a reference system, we establish a positive lower bound on the condensate density for continuous Bose systems with superstable two-body interactions and a finite gap in the one-particle excitations spectrum, i.e. we prove for the first time standard homogeneous Bose-Einstein condensation for such interacting systems

The stochastic limit in the analysis of the open BCS model

In this paper we show how the perturbative procedure known as {\em stochastic limit} may be useful in the analysis of the Open BCS model discussed by Buffet and Martin as a spin system interacting with a fermionic reservoir. In particular we show how the same values of the critical temperature and of the order parameters can be found with a significantly simpler approach

The Approximating Hamiltonian Method for the Imperfect Boson Gas

The pressure for the Imperfect (Mean Field) Boson gas can be derived in several ways. The aim of the present note is to provide a new method based on the Approximating Hamiltonian argument which is extremely simple and very general.Comment: 7 page

Equivalence of Bose-Einstein condensation and symmetry breaking

Based on a classic paper by Ginibre [Commun. Math. Phys. {\bf 8} 26 (1968)] it is shown that whenever Bogoliubov's approximation, that is, the replacement of a_0 and a_0^* by complex numbers in the Hamiltonian, asymptotically yields the right pressure, it also implies the asymptotic equality of ||^2/V and /V in symmetry breaking fields, irrespective of the existence or absence of Bose-Einstein condensation. Because the former was proved by Ginibre to hold for absolutely integrable superstable pair interactions, the latter is equally valid in this case. Apart from Ginibre's work, our proof uses only a simple convexity inequality due to Griffiths.Comment: An error in my summary of previous results (the definition of F') is corrected. The correction is to be done also in the PR

Many-particle quantum graphs and Bose-Einstein condensation

In this paper we propose quantum graphs as one-dimensional models with a complex topology to study Bose-Einstein condensation and phase transitions in a rigorous way. We fist investigate non-interacting many-particle systems on quantum graphs and provide a complete classification of systems that exhibit Bose-Einstein condensation. We then consider models of interacting particles that can be regarded as a generalisation of the well-known Tonks-Girardeau gas. Here our principal result is that no phase transitions occur in bosonic systems with repulsive hardcore interactions, indicating an absence of Bose-Einstein condensation