Spectral statistic for decaying random potentials

We consider Anderson model $H^{\omega}=-\Delta+V^{\omega}$ on $\ell^2(\mathbb{Z}^d)$ with decaying random potential. We study the point process $\xi^{\omega}_{L,\lambda}$ associated with eigenvalues of $H^{\omega}_{\Lambda_L}$, the retriction of $H^{\omega}$ to the finite cube $\Lambda_L$. Our result is that the weak limit points of $\{\xi^{\omega}_{L,\lambda}\}$ are poisson point processes as $L\to\infty$.Comment: This paper has been withdrawn by the author due to a crucial technical erro

Some Estimates Regarding Integrated density of States for Random Schr\"{o}dinger Operator with decaying Random Potentials

We investigate some bounds for the density of states in the pure point regime for the random Schr\"{o}dinger operators $H^{\omega}=-\Delta+\displaystyle\sum_{n\in\mathbb{Z}^d}a_nq_n(\omega)$, acting on $\ell^2(\mathbb{Z}^d)$, where $\{q_n\}$ are iid random variables and $a_n\simeq|n|^{-\alpha}~~\alpha>0$

Level Repulsion for a class of decaying random potentials

In this paper we consider the Anderson model with decaying randomness and show that statistics near the band edges in the absolutely continuous spectrum in dimensions $d \geq 3$ is independent of the randomness and agrees with that of the free part. We also consider the operators at small coupling and identify the length scales at which the statistics agrees with the free one in the limit when the coupling constant goes to zero

Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian

In this work, we focus on the multiplicity of singular spectrum for operators of the form $A^\omega=A+\sum_{n}\omega_n C_n$ on a separable Hilbert space $\mathcal{H}$, for a self-adjoint operator $A$ and a countable collection $\{C_n\}_{n}$ of non-negative finite rank operators. When $\{\omega_n\}_n$ are independent real random variables with absolutely continuous distributions, we show that the multiplicity of singular spectrum is almost surely bounded above by the maximum algebraic multiplicity of eigenvalues of $\sqrt{C_n}(A^\omega-z)^{-1}\sqrt{C_n}$ for all $n$ and almost all $(z,\omega)$. The result is optimal in the sense that there are operators where the bound is achieved. Using this, we also provide effective bounds on multiplicity of singular spectrum for some special cases.Comment: 31 pages, 4 figure

Spectral Statistics for one dimensional Anderson model with unbounded but decaying potential

In this work, we study the spectral statistics for Anderson model on $\ell^2(\mathbb{N})$ with decaying randomness whose single site distribution has unbounded support. Here we consider the operator $H^\omega$ given by $(H^\omega u)_n=u_{n+1}+u_{n-1}+a_n\omega_n u_n$, $a_n\sim n^{-\alpha}$ and $\{\omega_n\}$ are real i.i.d random variables following symmetric distribution $\mu$ with fat tail, i.e $\mu((-R,R)^c)<\frac{C}{R^\delta}$ for $R\gg 1$, for some constant $C$. In case of $\alpha-\frac{1}{\delta}>\frac{1}{2}$, we are able to show that the eigenvalue process in $(-2,2)$ is the clock process.Comment: 16 page

Eigenfunction Statistics for Anderson Model with H\"{o}lder continuous single site Potential

We consider random Schr\"{o}dinger operators on $\ell^2(\mathbb{Z}^d)$ when the distribution of single site potentials is $\alpha$-H\"{o}lder continuous ($0<\alpha\leq 1$). In localized regime we study the distribution of eigenfunctions simultaneously in space and energy. In a certain scaling limit we prove limits point are Poisson

Poisson Statistics for Anderson Model with Singular Randomness

In this work we consider the Anderson model on the $d$-dimensional lattice with the single site potential having singular distribution, mainly $\alpha$-H\"older continuous ones and show that the eigenvalue statistics is Poisson in the region of exponential localization

Spectral statistics of random Schr\"{o}dinger operator with growing potential

In this work we investigate the spectral statistics of random Schr\"{o}dinger operators $H^\omega=-\Delta+\sum_{n\in\mathbb{Z}^d}(1+|n|^\alpha)q_n(\omega)|\delta_n\rangle\langle\delta_n|$, $\alpha>0$ acting on $\ell^2(\mathbb{Z}^d)$ where $\{q_n\}_{n\in\mathbb{Z}^d}$ are i.i.d random variables distributed uniformly on $[0,1]$.Comment: 26 page

Phase separation in binary mixtures of active and passive particles

We study binary mixtures of small active and big passive athermal particles interacting via soft repulsive forces on a frictional substrate. Athermal self propelled particles are known to phase separate into a dense aggregate and a dilute gas-like phase at fairly low packing fractions. Known as {\emph {motility induced phase separation}}, this phenomenon governs the behaviour of binary mixtures for small to intermediate size ratios of the particle species. An effective attraction between passive particles, due to the surrounding active medium, leads to true phase separation for large size ratios and volume fractions of active particles. The effective interaction between active and passive particles can be attractive or repulsive at short range depending on the size ratio and volume fractions of the particles. This affects the clustering of passive particles. We find three distinct phases based on the spatial distribution of passive particles. The cluster size distribution of passive particles decays exponentially in the {\emph{homogeneous phase}}. It decays as a power law with an exponential cutoff in the {\emph{clustered phase}} and tends to a power law as the system approaches the transition to the {\emph{phase separated state}}. We present a phase diagram in the plane defined by the size ratio and volume fraction of passive particles.Comment: 9 pages, 11 figure

Regularity of the density of states of Random Schr\"odinger Operators

In this paper we solve a long standing open problem for Random Schr\"odinger operators on $L^2(\mathbb{R}^d)$ with i.i.d single site random potentials. We allow a large class of free operators, including magnetic potential, however our method of proof works only for the case when the random potentials satisfy a complete covering condition. We require that the supports of the random potentials cover $\mathbb{R}^d$ and the bump functions that appear in the random potentials form a partition of unity. For such models, we show that the Density of States (DOS) is $m$ times differentiable in the part of the spectrum where exponential localization is valid, if the single site distribution has compact support and has H\"older continuous $m+1$ st derivative. The required H\"older continuity depends on the fractional moment bounds satisfied by appropriate operator kernels. Our proof of the Random Schr\"odinger operator case is an extensions of our proof for Anderson type models on $\ell^2(\mathbb{G})$, $\mathbb{G}$ a countable set, with the property that the cardinality of the set of points at distance $N$ from any fixed point grows at some rate in $N^\alpha, \alpha >0$. This condition rules out the Bethe lattice, where our method of proof works but the degree of smoothness also depends on the localization length, a result we do not present here. Even for these models the random potentials need to satisfy a complete covering condition. The Anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition.Comment: 37 page