Delocalization transition in low energy excitation modes of vector spin glasses

We study the energy minima of the fully-connected mm-components vector spin glass model at zero temperature in an external magnetic field for m\ge 3m≥3. The model has a zero temperature transition from a paramagnetic phase at high field to a spin glass phase at low field. We study the eigenvalues and eigenvectors of the Hessian in the minima of the Hamiltonian. The spectrum is gapless both in the paramagnetic and in the spin glass phase, with a pseudo-gap behaving as \lambda^{m-1}λm−1 in the paramagnetic phase and as \sqrt{\lambda}λ at criticality and in the spin glass phase. Despite the long-range nature of the model, the eigenstates close to the edge of the spectrum display quasi-localization properties. We show that the paramagnetic to spin glass transition corresponds to delocalization of the edge eigenvectors. We solve the model by the cavity method in the thermodynamic limit. We also perform numerical minimization of the Hamiltonian for N\le 2048N≤2048 and compute the spectral properties, that show very strong corrections to the asymptotic scaling approaching the critical point.

On ℓp-Gaussian–Grothendieck Problem

For $p\geq 1$ and $(g_{ij})_{1\leq i,j\leq n}$ being a matrix of i.i.d. standard Gaussian entries, we study the $n$-limit of the $\ell _p$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $p=\infty $, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $1\leq p<2$ and $2<p<\infty .$ For the former, we compute the limit of the $\ell _p$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $n^{-1}$.

Acoustic-electrical testing of defects in the cement-sand and cement-glass model samples

A complex method of acoustic-electrical testing of defects in dielectric samples made from cement-sand and cement-glass mixtures is discussed. The paper reports the results of studies of changes in the parameters of electromagnetic responses and their spectra under pulsed deterministic acoustic excitation of model samples with defects in the form of solid-state inclusions. The results of mathematical calculations of the time variation in the stress-strain state induced in a defective dielectric model sample by deterministic acoustic pulse are presented. The relationship is shown between the parameters of the acoustic excitation and the electromagnetic response to the impact in a magnetic field. The study revealed that the specific electrical resistance of the cement-sand and cement-glass mixtures differs significantly. Excitation of electrical double layers by acoustic pulses causes an electromagnetic signal, parameters of which depend on the parameters of the acoustic impact and acoustic and electrical properties of the material. As a result, a reduced specific electrical resistance of the mixture increases its conductivity. The numerical calculation of the propagation of the deterministic acoustic pulse showed that its parameters change when it passes through a defect with acoustic impedance different from that of the mixture used.

Effects of Gravity on Foam Behavior in Roughened Model Fractures

Summary In this study, to investigate how gravity affects foam in open vertical fractures, we report foam experiments in three 1-m-long, 15-cm-wide glass-model fractures. Each fracture has a smooth wall and a roughened wall. Between the two walls is a slit-like channel representing a single geological fracture. Three model fractures (Models A, B, and C) share the same roughness and have different hydraulic apertures of 78, 98, and 128 µm, respectively. We conduct foam experiments by horizontal injection in the three model fractures placed horizontally and sideways (i.e., with the model fractures turned on their long side), and in Model A placed vertically with injection upward or downward. Direct imaging of the foam inside the model fracture is facilitated using a high-speed camera. We find that foam reaches local equilibrium (LE; where the rate of bubble generation equals that of bubble destruction) in horizontal-flow experiments in all three model fractures and in vertical-flow experiments in Model A. In fractures with a larger hydraulic aperture, foam is coarser because of less in-situ foam generation. In the vertical-flow experiments in Model A, we find that the properties of the foam are different in upward and downward flow. Compared with downward flooding, upward flooding creates a finer-texture foam, as sections near the inlet of this experiment are in a wetter state, which benefits in-situ foam generation. Moreover, less gas is trapped during upward flooding, as gravitational potential helps overcome the capillarity and moves bubbles upward. In the sideways-flow experiments, gravity segregation takes place. As a result, drier foam propagates along the top of the fractures and wetter foam along the bottom. The segregation is more significant in fractures with a larger hydraulic aperture. At foam quality 0.8, gas saturation is 27.7% greater at the top than the bottom for Model C, and 19.3% and 10.8% for Models B and A, respectively. Despite the gravity segregation in all three model fractures, water and gas are not completely segregated. All three model fractures thus represent a capillary transition zone, with greater segregation with increasing aperture. Our results suggest that the propagation of foam in vertical natural fractures meters tall and tens of meters long, with an aperture of hundreds of microns or greater, is problematic. Gravity segregation in foam would weaken its capacity in the field to maintain uniform flow and divert gas in a tall fracture over large distances.