Scaling Limits and Critical Behaviour of the 4-Dimensional n-Component |\varphi |^4 Spin Model

We consider the \(n\) -component \(|\varphi |^4\) spin model on \({\mathbb {Z}}^4\) , for all \(n \ge 1\) , with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent \(\frac{n+2}{n+8}\) for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for \(n =1,2,3\) ; double logarithmic scaling for \(n=4\) ; and is bounded when \(n>4\) . In addition, for the model defined on the \(4\) -dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the \(|\varphi |^4\) model.     

A novel methodological approach for δ18O analysis of sugars using gas chromatography-pyrolysis-isotope ratio mass spectrometry

Although the instrumental coupling of gas chromatography-pyrolysis-isotope ratio mass spectrometry (GC-Py-IRMS) for compound-specific δ¹⁸O analysis has been commercially available for more than a decade, this method has been hardly applied so far. Here we present the first GC-Py-IRMS δ¹⁸O results for trimethylsilyl-derivatives of plant sap-relevant sugars and a polyalcohol (glucose, fructose, sucrose, raffinose and pinitol). Particularly, we focus on sucrose, which is assimilated in leaves and which is the most important transport sugar in plants and hence of utmost relevance in plant physiology and paleoclimate studies. Replication measurements of sucrose standards and concentration series indicate that the GC-Py-IRMS δ¹⁸O measurements are not stable over time and that they are amount (area) dependent. We, therefore, suggest running sample batch replication measurements in alternation with standard concentration series of reference material. This allows for carrying out (i) a drift correction, (ii) a calibration against reference material and (iii) an amount (area) correction. Tests with ¹⁸O-enriched water do not provide any evidence for oxygen isotope exchange reactions affecting sucrose and raffinose. We present the first application of GC-Py-IRMS δ¹⁸O analysis for sucrose from needle extract (soluble carbohydrate) samples. The obtained δ¹⁸O_{sucrose/ Vienna Standard Mean Ocean Water (VSMOW)} values are more positive and vary in a wider range (32.1–40.1 ‰) than the δ¹⁸O_{bulk/ VSMOW} values (24.6–27.2 ‰). Furthermore, they are shown to depend on the climate parameters maximum day temperature, relative air humidity and cloud cover. These findings suggest that δ¹⁸O_sucrose of the investigated needles very sensitively reflects the climatically controlled evaporative ¹⁸O enrichment of leaf water and thus highlights the great potential of GC-Py-IRMS δ¹⁸O_sucrose analysis for plant physiology and paleoclimate studies.     

The raspberry model for protein-like particles: Ellipsoids and confinement in cylindrical pores

The study of protein mass transport via atomistic simulation requires time and length scales beyond the computational capabilities of modern computer systems. The raspberry model for colloidal particles in combination with the mesoscopic hydrodynamic method of lattice Boltzmann facilitates coarse-grained simulations that are on the order of microseconds and hundreds of nanometers for the study of diffusive transport of protein-like colloid particles. The raspberry model reproduces linearity in resistance to motion versus particle size and correct enhanced drag within cylindrical pores at off-center coordinates for spherical particles. Owing to the high aspect ratio of many proteins, ellipsoidal raspberry colloid particles were constructed and reproduced the geometric resistance factors of Perrin and of Happel and Brenner in the laboratory-frame and in the moving body-frame. Accurate body-frame rotations during diffusive motion have been captured for the first time using projections of displacements. The spatial discretization of the fluid leads to a renormalization of the hydrodynamic radius, however, the data describes a self-consistent hydrodynamic frame within this renormalized system.     

Phase transitions of nonlinear waves in quadratic waveguide arrays

We study two-color parametric nonlinear modes in waveguide arrays with a quadratic nonlinear response. We predict theoretically and observe experimentally a new type of phase transition manifested in an abrupt power-controlled change of the mode structure from unstaggered to staggered, due to the interplay of localization and synchronization in parametrically driven discrete systems.     

Local Semicircle Law for Random Regular Graphs

We consider random d‐regular graphs on N vertices, with degree d at least (log N)⁴. We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by Wigner's semicircle law, down to the optimal scale given by the typical eigenvalue spacing (up to a logarithmic correction). Aside from well‐known consequences for the local eigenvalue distribution, this result implies the complete (isotropic) delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity.© 2017 Wiley Periodicals, Inc.     

Structure and dynamics of alpha-lactalbumin adsorbed at a charged brush interface

We have studied the adsorption of alpha-lactalbumin at a planar poly(acrylic acid) (PAA) brush using neutron reflectometry (NR) and total internal reflection fluorescence (TIRF) spectroscopy. The PAA brush has been prepared by spin-coating silicon or quartz plates with a hydrophobic poly(styrene) film and by transferring the copolymer poly(styrene)-poly(acrylic acid) onto the modified surface. In the case of NR, the poly(styrene) film and the poly(styrene) chain ends of the copolymer were perdeuterated in order to generate a high contrast to the non-deuterated PAA brush. alpha-Lactalbumin was chosen as the model protein because it is a relatively small globular protein with a negative net charge at neutral pH-values, as chosen in the experiments. Thus, it is interacting with the PAA brush on the 'wrong' side of its isoelectric point. In addition, the effects of temperature on the volume fraction profile and the reorientational mobility of the protein within the PAA brush were determined. From the analysis of the NR data, it has been found that despite of its negative net charge, alpha-lactalbumin is penetrating into the PAA brush. Its volume fraction profile parallels that of the PAA brush, indicating an exclusive interaction between the protein and the PAA. No protein accumulation is found at the inner poly(styrene) or the outer solution interface of the PAA brush. When increasing the temperature from 20 to 40 degrees C, less protein is adsorbed, suggesting the presence of enthalpic interaction contributions. From the analysis of the TIRF data, a high degree of reorientational mobility of alpha-lactalbumin within a PAA brush can be inferred. The reorientational correlation time of alpha-lactalbumin labeled with the Alexa Fluor 488 dye was found to increase from 5.5 to 32 ns upon adsorption, which can well be explained by the higher viscosity inside the PAA brush. Overall, the results of this study quantify for the first time the molecular details of the unique interaction of a protein on the 'wrong' side of its isoelectric point with a planar charged brush interface. It is concluded that the high mobility of alpha-lactalbumin within a PAA brush can partially be understood by the presence of repulsive electrostatic interactions. There is no 'freezing' of the protein dynamics, which is a precondition for biological activity.