I have come across the following problem in an attempt to prove an entropy bound for large random Bernoulli-Toeplitz matrices (Conjecture 1 on p. 16 of this preprint by Clifford et al. 2015), which is essentially equivalent to Conjecture 2 in this earlier question.
Let $M$ be a $m{\times}n$ ($m<n$) Toeplitz (or circulant) matrix with random entries $\pm1$, and let $\gamma_0>0$ be some constant. Consider the following $n$-dimensional "Gaussian" sum: \begin{align*} \sum_{k \in \mathbb Z^n} {\rm e}^{-k^{\rm T}\big(I_n+\gamma_0 M^{\rm T}M\big)k} \ , \tag*{(1)} \end{align*} where $I_n$ denotes the $n{\times}n$ unit matrix. I want to prove that a certain upper bound on this sum (I'll explain the details below) is valid with nonzero probability.
Let me first motivate the type of bound I am looking for. Consider the spectral decomposition of the (symmetric) $n{\times}n$ matrix $M^{\rm T}M$. All eigenvectors are real and non-negative, and the eigenbasis is a family of orthonormal vectors in $\mathbb R^n$. It can be shown that for large $m$, $n$ (and $m/n$ small), the eigenvalues are concentrated around the value $n$; more precisely, their asymptotic distribution turns out to be sub-Gaussian with mean $n$ and variance $mn$. In contrast, very little seems to be known about the distribution of the eigenvectors, at least in this special case of random matrices of Toeplitz form, since they are more difficult to handle than random matrices with independent entries. However, one may reason heuristically as follows. Since $M^{\rm T}M$ is a random matrix, its eigenvector basis should in some sense be "randomly" oriented relative to the canonical basis of $\mathbb R^n$, and so one would expect that the sum behaves asymptotically (for large $m$ and $n$) like its average over all orientations in $\mathbb R^n$: \begin{align*} \sum_{k \in \mathbb Z^n} {\rm e}^{-k^{\rm T}\big(I_n+\gamma_0 M^{\rm T}M\big)k} & \sim \sum_{k \in \mathbb Z^n} \left\langle {\rm e}^{-k^{\rm T} R^{\rm T} \big( I_n+\gamma_0M^{\rm T}M \big) R\, k} \right\rangle_{\!R\in {\rm O}(n)} \ , \tag*{(2)} \end{align*} where $\big\langle {\cdot} \big\rangle_{R\in {\rm O}(n)}$ denotes an average over a uniformly (i.e., Haar) distributed orthogonal $n{\times}n$ random matrix $R$. Note that (2) will not hold individually for each summand (for instance, it obviously does not hold for any $k$ with $Mk=0$), but it may hold for the entire sum.
The interesting thing about (2) is that the right hand side only depends on the eigenvalue spectrum of $M^{\rm T}M$, not anymore on the eigenvectors, and therefore it is much easier to work with than the left hand side. Therefore, what I would like to do is to bound the l.h.s. of (2) by its r.h.s.. More precisely, what I need in the proof I mentioned is something like the following statement.
Conjecture:
Consider a sequence of $m{\times}n$ ($m$, $n$ monotonically increasing and $m<n$) random Toeplitz (or circulant) matrices $M$ whose independent entries are drawn i.u.d. from $\{-1,1\}$. Then there exist constants $\gamma_0,\, \gamma_1,\, C_1,\, C_2 > 0$ (with $C_2 < a_0$), $\tau \geq \frac12$, and $p \in\, ]0,1[$, all independent of $m$ and $n$, such that for large enough $m,n$ (and $\frac mn$ small enough) the following inequality holds with probability $\geq\!p$: \begin{align*} \sum_{\substack{k \in \mathbb Z^n \\ (|k|^2 \leq \tau n)}}\!\!\! {\rm e}^{-k^{\rm T}\big(I_n+\gamma_0 M^{\rm T}M\big)k} \leq C_1^m C_2^n \sum_{k \in \mathbb Z^n} \left\langle {\rm e}^{-k^{\rm T} R^{\rm T} \big( I_n+\gamma_1 M^{\rm T}M \big) R\, k} \right\rangle_{\!R\in {\rm O}(n)} \ . \tag*{(3)} \end{align*}
This statement is weaker than (2) in the following aspects: (i) the sum on the left has been truncated after a finite number of terms (which, however, increases with $n$), (ii) there are additional factors in the right expression which grow exponentially in $m$ and $n$, and (iii) the constant $\gamma_1$ on the right is now allowed to be different from (in particular, smaller than) $\gamma_0$ on the left.
I would be most grateful for any suggestion how to approach this conjecture, or for any reference to a relevant publication.
Added 2016-08-22: I have posted a reformulation of this question here.
