On criticality for competing influences of boundary and external field in the Ising model

Consider the Gibb's measures μΛ(1/h),-,s (defined below) of the Ising model, in a box Λ(l/h) in Zd with side length 1/h, with external field s and negative boundary condition at a temperature T B2(T), the limit is μ+. This says that the negative boundary conditon dominates in the limit when B B2(T). The question, then, is whether there exists a critical value B0 = B0(T) = B1(T) = B2(T) for all T B0. In the case of d = 2, this question was completely solved by Schonmann and Shlosman (1996), using large deviation results and techniques. For higher dimensions, Greenwood and Sun (1997) ([GS] hereafter) proved the criticality of a certain value B0 for all T B0. In [Sch], the main results are about the relaxation time of a stochastic Ising model in relation to an external field h. He shows that the relaxation time blows up when h ↘ 0 as exp(λ/hd-1). In fact he obtains upper and lower bounds for λ = λ(T), which are derived from his B1(T), B2 (T) and his estimate of the spectral gap of the generator of the evolution. One might hope to obtain a critical value of λ using Schonmann's methods and the critical value B0. This indeed again gives bounds for λ but not a critical value. A reason is that estimation of the spectral gap is involved

Reconstruction of primary vertices at the ATLAS experiment in Run 1 proton–proton collisions at the LHC

This paper presents the method and performance of primary vertex reconstruction in proton–proton collision data recorded by the ATLAS experiment during Run 1 of the LHC. The studies presented focus on data taken during 2012 at a centre-of-mass energy of √s=8 TeV. The performance has been measured as a function of the number of interactions per bunch crossing over a wide range, from one to seventy. The measurement of the position and size of the luminous region and its use as a constraint to improve the primary vertex resolution are discussed. A longitudinal vertex position resolution of about 30μm is achieved for events with high multiplicity of reconstructed tracks. The transverse position resolution is better than 20μm and is dominated by the precision on the size of the luminous region. An analytical model is proposed to describe the primary vertex reconstruction efficiency as a function of the number of interactions per bunch crossing and of the longitudinal size of the luminous region. Agreement between the data and the predictions of this model is better than 3% up to seventy interactions per bunch crossing