Overview
I am trying to bound from below the smallest singular value $\sigma_{n}$ of a sequence of symmetric $n$ by $n$ random matrices $M_{n}$ with dependant entries. In particular, I would like to find a lower bound of the form:
$\mathbb{P}[\sigma_{n} < \delta_{n}] < 1 - \epsilon_{n}$,
where $\delta_{n}$, $\epsilon_{n}$ don't go to 0 too quickly. Ideally, I would like to have $\delta_{n} > n^{-\log(n)^{C}}$ and $\epsilon_{n} > \frac{1}{\log(n)^{C}}$ for some $C < \infty$. My matrices $M_{n}$ do not have independent entries, though they do come with a moderate amount of independence built in. They also have all diagonal entries equal to 1, and all off-diagonal entries have mean 0 and are strictly between -1 and 1.
Related Research and Reference Request
- As far as I can tell from a brief survey, the closest nearby problem that has attracted a lot of attention is to show a much stronger bound for a very restricted class of matrices. For example, one might show
$\mathbb{P}[\sigma_{n} < 10 n^{-0.5}] \rightarrow 0$,
but only for matrices with i.i.d. Gaussian entries. The lecture notes http://www-personal.umich.edu/~romanv/papers/rv-ICM2010.pdf have a nice introduction to many relevant techniques for refining this type of estimate.
There has been some work on this problem for matrices with dependent entries. The most promising to me is in the paper http://arxiv.org/abs/math/0703307 of Tao and Vu, which rather casually gets good estimates for many matrices with dependent entries. Their results don't apply directly to my case, though they also are not very interested in matrices with dependent entries. The paper "Invertibility of random matrices: unitary and orthogonal perturbations" by Rudelson and Vershynin also deals with dependent entries, though again it is focused on getting good estimates for a small class of matrices.
I haven't seen anything that takes advantage of the random matrix of interest being a perturbation of a "nice" matrix. In my case, I know that the diagonals are all 1.
Any references to papers that get the types of relaxed bounds that I'm interested in would be very useful, even if they don't directly address my question below.
My Problem
The matrices $M_{n}$ I'm interested in are symmetric, have diagonal entries $M_{n}[i,i]=1$ and have super-diagonal entries
$M_{n}[i,j] = e_{i}^{t} (\prod_{k=i}^{j-1} Q_{i}) e_{j}$,
where $e_{i}, e_{j}$ are the $i$'th and $j$'th standard basis vectors and the matrices $Q_{k}$ are independent but not identically distributed elements of $SO(n)$. In particular, they are not distributed according to Haar measure and are often the identity.
The matrices $Q_{k}$ are of the form
$Q_{k} = \prod_{t=T_{k}+1}^{T_{k+1}} R(V_{t}, \theta_{t})$,
where $V_{t}$ is a sequence of $2$-dimensional planes (the distribution here is a little complicated), $\theta_{t}$ is an iid sequence of uniform $[0,2\pi]$ random variables that do not depend on the sequence $V_{t}$, and $R(V,\theta)$ is a rotation by angle $\theta$ in the plane $V$. I don't have a huge amount of control over the sequence $V_{t}$, though I know that it is large enough that the associated rotations can generate $SO(n)$.
Thanks for any suggestions!
