Delocalization and Diffusion Profile for Random Band Matrices

We consider Hermitian and symmetric random band matrices H = (h _xy ) in ${d\,\geqslant\,1}$ d ? 1 dimensions. The matrix entries h _xy , indexed by ${x,y \in (\mathbb{Z}/L\mathbb{Z})^d}$ x , y ∈ ( Z / L Z ) d , are independent, centred random variables with variances ${s_{xy} = \mathbb{E} |h_{xy}|^2}$ s x y = E | h x y | 2 . We assume that s _xy is negligible if |x ? y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if ${W\gg L^{4/5}}$ W ? L 4 / 5 . We also show that the magnitude of the matrix entries ${|{G_{xy}}|^2}$ | G x y | 2 of the resolvent ${G=G(z)=(H-z)^{-1}}$ G = G ( z ) = ( H - z ) - 1 is self-averaging and we compute ${\mathbb{E} |{G_{xy}}|^2}$ E | G x y | 2 . We show that, as ${L\to\infty}$ L → ∞ and ${W\gg L^{4/5}}$ W ? L 4 / 5 , the behaviour of ${\mathbb{E} |G_{xy}|^2}$ E | G x y | 2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.     

A Strong Central Limit Theorem for a Class of Random Surfaces

This paper is concerned with d = 2 dimensional lattice field models with action ${V(\nabla\phi(\cdot))}$ , where ${V : \mathbf{R}^d \rightarrow \mathbf{R}}$ is a uniformly convex function. The fluctuations of the variable ${\phi(0) - \phi(x)}$ are studied for large |x| via the generating function given by ${g(x, \mu) = \ln \langle e^{\mu(\phi(0) - \phi(x))}\rangle_{A}}$ . In two dimensions ${g'' (x, \mu) = \partial^2g(x, \mu)/\partial\mu^2}$ is proportional to ${\ln\vert x\vert}$ . The main result of this paper is a bound on ${g''' (x, \mu) = \partial^3 g(x, \mu)/\partial \mu^3}$ which is uniform in ${\vert x \vert}$ for a class of convex V. The proof uses integration by parts following Helffer–Sjöstrand and Witten, and relies on estimates of singular integral operators on weighted Hilbert spaces.     

Self-Avoiding Walk is Sub-Ballistic

We prove that self-avoiding walk on ${\mathbb{Z}^d}$ is sub-ballistic in any dimension d ≥ 2. That is, writing ${\| u \|}$ for the Euclidean norm of ${u \in \mathbb{Z}^d}$ , and ${\mathsf{P_{SAW}}_n}$ for the uniform measure on self-avoiding walks ${\gamma : \{0, \ldots, n\} \to \mathbb{Z}^d}$ for which γ ₀ = 0, we show that, for each v > 0, there exists ${\varepsilon > 0}$ such that, for each ${n \in \mathbb{N}, \mathsf{P_{SAW}}_n \big( {\rm max}\big\{\| \gamma_k \| : 0 \leq k \leq n\big\} \geq vn \big) \leq e^{-\varepsilon n}}$ .     

Magnetotransport properties in \text{ Ni }_{81}\text{ Fe }_{19}\text{/Zr } multi-layers

In this work, we present a study of the magneto transport properties in magnetic multilayered structure $\text{ Ni }_{81}\text{ Fe }_{19}\text{/Zr }$ Ni 81 Fe 19 /Zr . The magnetic $(\text{ Ni }_{81}\text{ Fe }_{19})$ ( Ni 81 Fe 19 ) and non magnetic (Zr) layer thickness $(\mathbf{t}_\mathbf{NiFe}, \mathbf{t}_\mathbf{zr})$ ( t NiFe , t zr ) effects on the magneto resistance (MR) are discussed theoretically in the framework of the Johnson–Camley semi classical approach based on the Boltzmann transport equation. A comparison between calculated and measured MR is obtained. The observed MR ratio oscillates for Zr layer thickness with an average period of 7Å. A generally weak $\text{ MR }(\text{ t }_{\mathrm{NiFe}})$ MR ( t NiFe ) ratio for fixed $\mathbf{t}_\mathbf{zr}$ t zr is obtained and it shows a maxima peak of the MR with a value of 1.8 % located at $\mathbf{t}_\mathbf{NiFe}= 80$ t NiFe = 80 Å.     

Long-time Asymptotic for the Derivative Nonlinear Schrödinger Equation with Step-like Initial Value

We study long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation i q t + q xx ? i q 2 q ? x + 1 2 | q | 4 q = 0 $$iq_{t}+q_{xx}-iq^{2}\bar{q}_{x}+\frac{1}{2}|q|^{4}{q}=0 $$ with step-like initial data q ( x , 0 ) = 0 $q(x,0)=0$ for x ≤ 0 $x \leqslant 0$ and q ( x , 0 ) = A e ? 2 iBx $q(x,0)=A\mathrm {e}^{-2iBx}$ for x > 0 $x>0$ , where A > 0 $A>0$ and B ∈ ? $B\in \mathbb R$ are constants. We show that there are three regions in the half-plane { ( x , t ) | ? ∞ < x < ∞ , t > 0 } $\{(x,t) | -\infty <x<\infty , t>0\}$ , on which the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for x > ? 4 tB $x>-4tB$ , a plane wave region: x > ? 4 t B + 2 A 2 B + A 2 4 $x<-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )$ , an elliptic region: ? 4 t B + 2 A 2 B + A 2 4 > x > ? 4 tB $-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )<x<-4tB$ . Our main tools include asymptotic analysis, matrix Riemann-Hilbert problem and Deift-Zhou steepest descent method.     

Global Existence for the Critical Dissipative Surface Quasi-Geostrophic Equation

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in ${\mathbb{R}^2}$ R 2 . Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data θ ₀ liying in the space ${\Lambda^{s} (\dot{H}^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$ Λ s ( H ˙ u l o c s ( R 2 ) ) ∩ L ∞ ( R 2 ) the critical (SQG) has a global weak solution in time for 1/2 <  s <  1. Our proof is based on an energy inequality verified by the equation ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? which is nothing but the (SQG) equation with truncated and regularized initial data. By classical compactness arguments, we show that we are able to pass to the limit ( ${R \rightarrow \infty}$ R → ∞ , ${\epsilon \rightarrow 0}$ ? → 0 ) in ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? and that the limit solution has the desired regularity.     

Regularity for Eigenfunctions of Schr?dinger Operators

We prove a regularity result in weighted Sobolev (or Babu?ka?CKondratiev) spaces for the eigenfunctions of certain Schr?dinger-type operators. Our results apply, in particular, to a non-relativistic Schr?dinger operator of an N-electron atom in the fixed nucleus approximation. More precisely, let ${\mathcal{K}_{a}^{m}(\mathbb{R}^{3N},r_S)}$ be the weighted Sobolev space obtained by blowing up the set of singular points of the potential ${V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}}$ , ${x \in \mathbb{R}^{3N}}$ , ${b_j, c_{ij} \in \mathbb{R}}$ . If ${u \in L^2(\mathbb{R}^{3N})}$ satisfies ${(-\Delta + V) u = \lambda u}$ in distribution sense, then ${u \in \mathcal{K}_{a}^{m}}$ for all ${m \in \mathbb{Z}_+}$ and all a ?? 0. Our result extends to the case when b _j and c _ij are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a?<?3/2.     

Investigation on an ultra-compact 1\times 2 polymer electro-optic switch using cross-coupling 2N+1 vertical-turning serial-coupled microrings

Generic model and thorough investigation are proposed for a novel $1\times 2$ 1 × 2 polymer electro-optic (EO) switch based on one-group $2N+1$ 2 N + 1 vertical-turning serial-coupled microrings. For realizing boxlike flat spectrum as well as low crosstalk and insertion loss, resonance order and coupling gaps are optimized. The MRR switches with $N \ge 1$ N ≥ 1 reveal favorable boxlike spectrum as when compared with the simple device with only one microring ( $N = 0$ N = 0 ). For obtaining $<-30\,\text{ dB }$ < - 30 dB crosstalk under through-state, the dependency of switching voltage on $N$ N is determined as $7.19 \times \text{ exp }(-N/0.72) + 1.72\,(\text{ V })$ 7.19 × exp ( - N / 0.72 ) + 1.72 ( V ) . Under the operation voltages of 0 V (drop state) and the predicted switching voltages (through state), the device performances are analyzed, and $1 \le N \le 10$ 1 ≤ N ≤ 10 is required for dropping the insertion loss (drop state) below 10 dB. The crosstalk of the ten devices ( $N = 1-10$ N = 1 - 10 ) are $< -19.5\,\text{ dB }$ < - 19.5 dB under drop state and $< -28.7\,\text{ dB }$ < - 28.7 dB under through state, and the insertion losses of the devices ( $N = 1-10$ N = 1 - 10 ) are $< 9.715\,\text{ dB }$ < 9.715 dB under drop state and $< 1.573\,\text{ dB }$ < 1.573 dB under through state. The device also has ultra-compact footprint size of only 0.33–1.06 mm, which is only 1/10–1/3 of those of our previously reported polymer EO switches based on directional coupler or Mach–Zehnder interferometer structures. Therefore, the proposed device is capable of highly integration onto optical networks-on-chip.     

Random Parking, Euclidean Functionals, and Rubber Elasticity

We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions S of subsets of ${\mathbb{R}^d}$ and of point sets that are (almost) subadditive in their first variable. Denoting by ξ the random parking measure in ${\mathbb{R}^d}$ , and by ξ ^R the random parking measure in the cube Q _R =  (?R, R) ^d , we show, under some natural assumptions on S, that there exists a constant ${\overline{S} \in \mathbb{R}}$ such that $$\lim_{R \to +\infty} \frac{S(Q_R, \xi)}{|Q_R|} \, = \, \lim_{R \to +\infty} \frac{S(Q_R, \xi^{R})}{|Q_R|} \, = \, \overline{S}$$ almost surely. If ${\zeta \mapsto S(Q_R, \zeta)}$ is the counting measure of ${\zeta}$ in Q _R , then we retrieve the result by the second author on the existence of the jamming limit. The present work generalizes this result to a wide class of (almost) subadditive functions. In particular, classical Euclidean optimization problems as well as the discrete model for rubber previously studied by Alicandro, Cicalese, and the first author enter this class of functions. In the case of rubber elasticity, this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit, as well as a generalization to stochastic networks generated on bounded sets.     

An interacting dark energy model in a non-flat universe

In this paper, an interacting dark energy model in a non-flat universe is studied, with taking interaction form $C=\alpha H\rho _{de}$ C = α H ρ d e . And in this study a property for the mysterious dark energy is aforehand assumed, i.e. its equation of state $w_{\Lambda }=-1$ w Λ = - 1 . After several derivations, a power-law form of dark energy density is obtained $\rho _{\Lambda } \propto a^{-\alpha }$ ρ Λ ∝ a - α , here $a$ a is the cosmic scale factor, $\alpha $ α is a constant parameter introducing to describe the interaction strength and the evolution of dark energy. By comparing with the current cosmic observations, the combined constraints on the parameter $\alpha $ α is investigated in a non-flat universe. For the used data they include: the Union2 data of type Ia supernova, the Hubble data at different redshifts including several new published datapoints, the baryon acoustic oscillation data, the cosmic microwave background data, and the observational data from cluster X-ray gas mass fraction. The constraint results on model parameters are $\Omega _{K}=0.0024\,(\pm 0.0053)^{+0.0052+0.0105}_{-0.0052-0.0103}, \alpha =-0.030\,(\pm 0.042)^{+0.041+0.079}_{-0.042-0.085}$ Ω K = 0.0024 ( ± 0.0053 ) - 0.0052 - 0.0103 + 0.0052 + 0.0105 , α = - 0.030 ( ± 0.042 ) - 0.042 - 0.085 + 0.041 + 0.079 and $\Omega _{0m}=0.282\,(\pm 0.011)^{+0.011+0.023}_{-0.011-0.022}$ Ω 0 m = 0.282 ( ± 0.011 ) - 0.011 - 0.022 + 0.011 + 0.023 . According to the constraint results, it is shown that small constraint values of $\alpha $ α indicate that the strength of interaction is weak, and at $1\sigma $ 1 σ confidence level the non-interacting cosmological constant model can not be excluded.     

Error Analysis on Photonic Qubit Rotations Implemented by Wave Plates

Optical Poincare sphere rotations $e^{-i\theta\sigma_{x}/2}$ , $e^{-i\theta\sigma_{y}/2}$ and $e^{-i\theta\sigma_{z}/2}$ can be realized by wave-plate combinations. Errors due to combinations with non-ideal wave plates are discussed for three specific combinations (θ=π) by trace distance. The result shows that different settings of combinations affect trace distance: (i) trace distance for $e^{-i\pi\sigma_{x}/2}$ equals that for $e^{-i\pi\sigma_{z}/2}$ , but both of them are smaller than that for $e^{-i\pi\sigma_{y}/2}$ , when optics-axis random errors are considered; (ii) trace distance for $e^{-i\pi\sigma_{x}/2}$ also equals that for $e^{-i\pi\sigma_{z}/2}$ , but both of them are larger than that for $e^{-i\pi\sigma_{y}/2}$ , when phase-shift random errors are considered. The method outlined in this paper is general and is useful to analyze other combinations.     

One-Dimensional Disordered Quantum Mechanics and Sinai Diffusion with Random Absorbers

We study the one-dimensional Schrödinger equation with a disordered potential of the form $$\begin{aligned} V (x) = \phi (x)^2+\phi '(x) + \kappa (x) \end{aligned}$$ where $\phi (x)$ is a Gaussian white noise with mean $\mu g$ and variance $g$ , and $\kappa (x)$ is a random superposition of delta functions distributed uniformly on the real line with mean density $\rho $ and mean strength $v$ . Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers: $\phi (x)$ models the force field acting on the diffusing particle and $\kappa (x)$ models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent $ \varOmega (E) = \gamma (E) - \mathrm{i}\pi N(E) $ , where $N$ is the integrated density of states per unit length and $\gamma $ the reciprocal of the localisation length. By using the continuous version of the Dyson–Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean $v=2g$ . Building on this result, we then solve the general case— in the low-energy limit— in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour $$\begin{aligned} N(E) \underset{E\rightarrow 0+}{\sim } E^\nu \quad \hbox {where } \quad \nu =\sqrt{\mu ^2+2\rho /g}. \end{aligned}$$ This confirms and extends several results obtained previously by approximate methods.     

Existence of a Non-Averaging Regime for the Self-Avoiding Walk on a High-Dimensional Infinite Percolation Cluster

Let $Z_N$ be the number of self-avoiding paths of length $N$ starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on ${\mathbb Z} ^d$ with parameter $p>p_c({\mathbb Z} ^d)$ . The object of this paper is to study the connective constant of the dilute lattice $\limsup _{N\rightarrow \infty } Z_N^{1/N}$ , which is a non-random quantity. We want to investigate if the inequality $\limsup _{N\rightarrow \infty } (Z_N)^{1/N} \le \lim _{N\rightarrow \infty } {\mathbb E} [Z_N]^{1/N}$ obtained with the Borel–Cantelli Lemma is strict or not. In other words, we want to know if the quenched and annealed versions of the connective constant are equal. On a heuristic level, this indicates whether or not localization of the trajectories occurs. We prove that when $d$ is sufficiently large there exists $p^{(2)}_c>p_c$ such that the inequality is strict for $p\in (p_c,p^{(2)}_c)$ .     

Universality of the Local Regime for the Block Band Matrices with a Finite Number of Blocks

We consider the block band matrices, i.e. the Hermitian matrices $H_N$ , $N=|\Lambda |W$ with elements $H_{jk,\alpha \beta }$ , where $j,k \in \Lambda =[1,m]^d\cap \mathbb {Z}^d$ (they parameterize the lattice sites) and $\alpha , \beta = 1,\ldots , W$ (they parameterize the orbitals on each site). The entries $H_{jk,\alpha \beta }$ are random Gaussian variables with mean zero such that $\langle H_{j_1k_1,\alpha _1\beta _1}H_{j_2k_2,\alpha _2\beta _2}\rangle =\delta _{j_1k_2}\delta _{j_2k_1} \delta _{\alpha _1\beta _2}\delta _{\beta _1\alpha _2} J_{j_1k_1},$ where $J=1/W+\alpha \Delta /W$ , $\alpha < 1/4d$ . This matrices are the special case of Wegner’s $W$ -orbital models. Assuming that the number of sites $|\Lambda |$ is finite, we prove universality of the local eigenvalue statistics of $H_N$ for the energies $|\lambda _0|< \sqrt{2}$ .     

Almost Commuting Unitary Matrices Related to Time Reversal

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either ${{\bf M}_n(\mathbb{A})}$ M n ( A ) for ${\mathbb{A} = \mathbb{R}}$ A = R , ${\mathbb{A} = \mathbb{C}}$ A = C or ${\mathbb{A} = \mathbb{H}}$ A = H , the algebra of quaternions. There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple.We consider two-dimensional geometries and so this obstruction lives in ${KO_{-2}(\mathbb{A})}$ K O - 2 ( A ) . This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.     

Truncated Connectivities in a Highly Supercritical Anisotropic Percolation Model

We consider an anisotropic bond percolation model on $\mathbb{Z}^{2}$ , with p=(p _h ,p _v )∈[0,1]², p _v >p _h , and declare each horizontal (respectively vertical) edge of $\mathbb{Z}^{2}$ to be open with probability p _h (respectively p _v ), and otherwise closed, independently of all other edges. Let $x=(x_{1},x_{2}) \in\mathbb{Z}^{2}$ with 0<x ₁<x ₂, and $x'=(x_{2},x_{1})\in\mathbb{Z}^{2}$ . It is natural to ask how the two point connectivity function $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $\mathbb{P}_{\mathbf{p}}(\{0\leftrightarrow x\})>\mathbb{P}_{\mathbf {p}}(\{0\leftrightarrow x'\})$ . In this note we give an affirmative answer in the highly supercritical regime.     

Local structure of Fe-doped In 2 O 3 films investigated by X-ray absorption fine structure spectroscopy

$(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ $(x=0.07, 0.09, 0.16, 0.22, 0.31)$ films were deposited on Si (100) substrates by RF-magnetron sputtering technique. The influence of Fe doping on the local structure of films was investigated by X-ray absorption spectroscopy (XAS) at Fe K-edge and L-edge. For the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ films with $x=0.07, 0.09 \mbox{ and } 0.16$ , Fe ions dissolve into $\mathrm{In}_{2}\mathrm{O}_{3}$ and substitute for $\mathrm{In}^{3+}$ sites with a mixed-valence state ( $\mathrm{Fe}^{2+}/\mathrm{Fe}^{3+}$ ) of Fe ions. However, a secondary phase of Fe metal clusters is formed in the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ films with $x=0.22 \mbox{ and } 0.31$ . The qualitative analyses of Fe-K edge extended X-ray absorption fine structure (EXAFS) reveal that the Fe–O bond length shortens and the corresponding Debye–Waller factor ( $\sigma^{2}$ ) increases with the increase of Fe concentration, indicating the relaxation of oxygen environment of Fe ions upon substitution. The anomalously large structural disorder and very short Fe–O distance are also observed in the films with high Fe concentration. Linear combination fittings at Fe L-edge further confirm the coexistence of $\mathrm{Fe}^{2+}$ and $\mathrm{Fe}^{3+}$ with a ratio of ${\sim}3:2$ ( $\mathrm{Fe}^{2+}: \mathrm{Fe}^{3+}$ ) for the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ film with $x=0.16$ . However, a significant fraction ( ${\sim}40~\mbox{at\%}$ ) of the Fe metal clusters is found in the $(\mathrm{In}_{1-x}\mathrm{Fe}_{x})_{2}\mathrm{O}_{3}$ film with $x=0.31$ .     

Finite W-Superalgebras and Truncated Super Yangians

Let ${Y_{m|n}^{\ell}}$ be the super Yangian of general linear Lie superalgebra for ${\mathfrak{gl}_{m|n}}$ . Let ${e \in \mathfrak{gl}_{m\ell|n\ell}}$ be a “rectangular” nilpotent element and ${\mathcal{W}_e}$ be the finite W-superalgebra associated to e. We show that ${Y_{m|n}^{\ell}}$ is isomorphic to ${\mathcal{W}_e}$ .     

Analysis of the {3\over 2}^{+} heavy and doubly heavy baryon states with QCD sum rules

In this article, we study the ${3\over 2}^{+}$ heavy and doubly heavy baryon states $\varXi^{*}_{cc}$ , $\varOmega^{*}_{cc}$ , $\varXi^{*}_{bb}$ , $\varOmega^{*}_{bb}$ , $\varSigma_{c}^{*}$ , $\varXi_{c}^{*}$ , $\varOmega_{c}^{*}$ , $\varSigma_{b}^{*}$ , $\varXi_{b}^{*}$ and $\varOmega_{b}^{*}$ by subtracting the contributions from the corresponding ${3\over 2}^{-}$ heavy and doubly heavy baryon states with the QCD sum rules, and we make reasonable predictions for their masses.