Let $P_n^1,\dots , P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, set $S_n^d:=\sum _{\ell =1}^d P_n^\ell $. We show that if $\log ^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d} $ converges weakly to circular law in probability as $n \to \infty

We show that the empirical eigenvalue measure for sum of $d$ independent Haar distributed $n$-dimensional unitary matrices, converge $n \rightarrow \infty$ to Brown free operators. The same applies orthogonal matrices. As a byproduct our approach, we relax requirement uniformly bounded imaginary part Stieltjes transform $T_n$ is made in [Guionnet, Krishnapur, Zeitouni; Theorem 1].

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript upper N"> <mml:semantics> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">T_N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote an N times <mml:mrow> <mml:mo>×</mml:mo> </mml:mrow> encoding="application/x-tex">N\times N</mml:annotation> Toeplitz matrix with finite,...

For a class of sparse random matrices the form $A_{n}=(\xi_{i,j}\delta_{i,j})_{i,j=1}^{n}$, where $\{\xi_{i,j}\}$ are i.i.d. centered sub-Gaussian variables unit variance, and $\{\delta_{i,j}\}$ Bernoulli taking value $1$ with probability $p_{n}$, we prove that empirical spectral distribution $A_{n}/\sqrt{np_{n}}$ converges weakly to circular law, in probability, for all $p_{n}$ such $p_{n}=\omega({\log^{2}n}/{n})$. Additionally if satisfies inequality $np_{n}>\exp(c\sqrt{\log n})$ some...

We consider the spectrum of additive, polynomially vanishing random perturbations deterministic matrices, as follows. Let $M_{N}$ be a $N\times N$ matrix, and let $G_{N}$ complex Ginibre matrix. matrix ${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$ , where $\unicode[STIX]{x1D6FE}&gt;1/2$ . With $L_{N}$ empirical measure eigenvalues ${\mathcal{M}}_{N}$ we provide general equivalence theorem that ties to singular values $z-M_{N}$ with $z\in \mathbb{C}$ then compute limit when is an...

The topological quantum field-effect transition in buckled two-dimensional-Xenes can potentially enable subthermionic transistor operation coupled with a dissipationless on-state conduction. We investigate realistic device structures that exploit the between phase and band-insulator phase. find previously considered dual-gate structure is disadvantageous, leading to near doubling of subthreshold swing. However, we identify single-gate strategy capable overcoming thermionic limit at cost...

Except the Toeplitz and Hankel matrices, common patterned matrices for which limiting spectral distribution (LSD) are known to exist, share a property--the number of times each random variable appears in matrix is (more or less) same across variables. Thus it seems natural ask what happens spectrum when entry scaled by square root that instead uniform scaling $n^{-1/2}$. We show LSD these balanced exist derive integral formulae moments limit distribution. Curiously, not clear if define unique

We investigate the effects of nitrogen passivation on band structure and density states in zigzag graphene nanoribbon (zzGNR) using first principle quantum mechanical simulations. The results show that edge termination zzGNR produces a bandgap (~0.7eV) around Fermi level. analyze Bloch functions projected for understanding origin bandgap. Based these findings, we propose nitrogen-passivated FET having n-type electrodes p-type scattering region boron doping, respectively. simulate its...

We show that the empirical spectral distribution (ESD) of sample autocovariance matrix (ACVM) converges as dimension increases, when time series is a linear process with reasonable restriction on coefficients. The limit does not depend underlying driving i.i.d. sequence and its support unbounded. This coincide theoretical ACVM. However, it so if we consider suitably tapered version For banded ACVM has unbounded long number non-zero diagonals in proportion to bounded away from zero. If this...

We analyze the electric field driven topological effect transition on 2D-xene materials with addition of momentum relaxation effects, in order to account for dephasing processes. The between quantum spin Hall phase and valley is analyzed detail using Keldysh non-equilibrium Green's function technique inclusion relaxation, within self-consistent Born approximation. Details applied are elucidated ON-OFF characteristics emphasis transport properties along tomography current carrying edge...

Gate patterning on semiconductors is routinely used to electrostatically restrict electron movement into reduced dimensions. At cryogenic temperatures, where most studies are carried out, differential thermal contraction between the patterned gate and semiconductor often lead an appreciable strain modulation. The impact of such modulated conductive channel buried in a has long been recognized, but measuring its magnitude variation rather challenging. Here we present way measure that...

For a ‐regular connected graph H the problem of determining upper tail large deviation for number copies in , an Erdős‐Rényi on n vertices with edge probability p has generated significant interest. and where is event believed to occur due presence localized structures. In this regime that exceeds its expectation by constant factor predicted hold at speed rate function conjectured be given solution mean‐field variational problem. After series developments recent years, covering progressively...