Isoperimetry in two-dimensional percolation

We consider the unique infinite connected component of supercritical bond percolation on the square lattice and study the geometric properties of isoperimetric sets, i.e., sets with minimal boundary for a given volume. For almost every realization of the infinite connected component we prove that, as the volume of the isoperimetric set tends to infinity, its asymptotic shape can be characterized by an isoperimetric problem in the plane with respect to a particular norm. As an application we then show that the anchored isoperimetric profile with respect to a given point as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the plane.Comment: 40 pages, 2 figs; version to appear in Commun. Pure Appl. Mat

Orbital order in classical models of transition-metal compounds

We study the classical 120-degree and related orbital models. These are the classical limits of quantum models which describe the interactions among orbitals of transition-metal compounds. We demonstrate that at low temperatures these models exhibit a long-range order which arises via an "order by disorder" mechanism. This strongly indicates that there is orbital ordering in the quantum version of these models, notwithstanding recent rigorous results on the absence of spin order in these systems.Comment: 7 pages, 1 eps fi

The Two-Screen Measurement Setup to Indirectly Measure Proton Beam Self-Modulation in AWAKE

The goal of the first phase of the AWAKE \cite{AWAKE1,AWAKE2} experiment at CERN is to measure the self-modulation \cite{SMI} of the $\sigma_z = 12\,\rm{cm}$ long SPS proton bunch into microbunches after traversing $10\,\rm{m}$ of plasma with a plasma density of $n_{pe}=7\times10^{14}\,\rm{electrons/cm}^3$. The two screen measurement setup \cite{Turner2016} is a proton beam diagnostic that can indirectly prove the successful development of the self-modulation of the proton beam by imaging protons that got defocused by the transverse plasma wakefields after passing through the plasma, at two locations downstream the end of the plasma. This article describes the design and realization of the two screen measurement setup integrated in the AWAKE experiment. We discuss the performance and background response of the system based on measurements performed with an unmodulated Gaussian SPS proton bunch during the AWAKE beam commissioning in September and October 2016. We show that the system is fully commissioned and adapted to eventually image the full profile of a self-modulated SPS proton bunch in a single shot measurement during the first phase of the AWAKE experiment.Comment: 5 pages 8 figure

Indirect Self-Modulation Instability Measurement Concept for the AWAKE Proton Beam

AWAKE, the Advanced Proton-Driven Plasma Wakefield Acceleration Experiment, is a proof-of-principle R&D experiment at CERN using a 400 GeV/c proton beam from the CERN SPS (longitudinal beam size sigma_z = 12 cm) which will be sent into a 10 m long plasma section with a nominal density of approx. 7x10^14 atoms/cm3 (plasma wavelength lambda_p = 1.2mm). In this paper we show that by measuring the time integrated transverse profile of the proton bunch at two locations downstream of the AWAKE plasma, information about the occurrence of the self-modulation instability (SMI) can be inferred. In particular we show that measuring defocused protons with an angle of 1 mrad corresponds to having electric fields in the order of GV/m and fully developed self-modulation of the proton bunch. Additionally, by measuring the defocused beam edge of the self-modulated bunch, information about the growth rate of the instability can be extracted. If hosing instability occurs, it could be detected by measuring a non-uniform defocused beam shape with changing radius. Using a 1 mm thick Chromox scintillation screen for imaging of the self-modulated proton bunch, an edge resolution of 0.6 mm and hence a SMI saturation point resolution of 1.2 m can be achieved.Comment: 4 pages, 4 figures, EAAC conference proceeding

Mean-field driven first-order phase transitions in systems with long-range interactions

We consider a class of spin systems on $\Z^d$ with vector valued spins (\bS_x) that interact via the pair-potentials J_{x,y} \bS_x\cdot\bS_y. The interactions are generally spread-out in the sense that the $J_{x,y}$'s exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions $d\ge3$, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions $d=1,2$ for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique ``state,'' then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure.Comment: 57 pages; uses a (modified) jstatphys class fil

Optimal designs for rational function regression

We consider optimal non-sequential designs for a large class of (linear and nonlinear) regression models involving polynomials and rational functions with heteroscedastic noise also given by a polynomial or rational weight function. The proposed method treats D-, E-, A-, and $\Phi_p$-optimal designs in a unified manner, and generates a polynomial whose zeros are the support points of the optimal approximate design, generalizing a number of previously known results of the same flavor. The method is based on a mathematical optimization model that can incorporate various criteria of optimality and can be solved efficiently by well established numerical optimization methods. In contrast to previous optimization-based methods proposed for similar design problems, it also has theoretical guarantee of its algorithmic efficiency; in fact, the running times of all numerical examples considered in the paper are negligible. The stability of the method is demonstrated in an example involving high degree polynomials. After discussing linear models, applications for finding locally optimal designs for nonlinear regression models involving rational functions are presented, then extensions to robust regression designs, and trigonometric regression are shown. As a corollary, an upper bound on the size of the support set of the minimally-supported optimal designs is also found. The method is of considerable practical importance, with the potential for instance to impact design software development. Further study of the optimality conditions of the main optimization model might also yield new theoretical insights.Comment: 25 pages. Previous version updated with more details in the theory and additional example

New strategies to measure intracellular sodium concentrations

Fluorescent ion indicators are widely used to measure ion concentrations in living cells. However, despite considerable efforts in synthesizing new compounds, no ratiometric sodium indicator is available that can be excited at visible wavelengths. Ratiometric indicators have an advantage in that measured fluorescence intensities can be corrected for fluctuations of the indicator concentration and the illumination intensity, which is not possible when non-ratiometric indicators are used. One way to circumvent this problem is to measure fluorescence lifetimes, which are independent of these factors. Another way to overcome the disadvantages of a non-ratiometric indicator dye is to embed it, together with a reference dye, into nanoparticles. By relating the indicator fluorescence to the fluorescence of the reference dye, inhomogeneities in the nanosensor concentration or the illumination intensity can be cancelled out reliably. In this study we compare the benefits and drawbacks of these approaches. © 2010 Copyright SPIE - The International Society for Optical Engineering

Annealed lower tails for the energy of a polymer

We consider the energy of a randomly charged polymer. We assume that only charges on the same site interact pairwise. We study the lower tails of the energy, when averaged over both randomness, in dimension three or more. As a corollary, we obtain the correct temperature-scale for the Gibbs measure.Comment: 27 page