Abstract
Lyapunov exponents characterize the asymptotic behavior of long matrix products. In this work we introduce a new technique that yields quantitative lower bounds on the top Lyapunov exponent in terms of an efficiently computable matrix sum in ergodic situations. Our approach rests on two results from matrix analysis—the n-matrix extension of the Golden–Thompson inequality and an effective version of the Avalanche Principle. While applications of this method are currently restricted to uniformly hyperbolic cocycles, we include specific examples of ergodic Schrödinger cocycles of polymer type for which outside of the spectrum our bounds are substantially stronger than the standard Combes–Thomas estimates. We also show that these techniques yield short proofs of quantitative stability results for the top Lyapunov exponent which are known from more dynamical approaches. We also discuss the problem of finding stable bounds on the Lyapunov exponent for almost-commuting matrices.
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Notes
For experts on Schrödinger operators, we note that the polymer models we construct here have spectrum along roughly a \(\frac{1}{p}\)-grid where p is a large parameter. Our bounds give order-1 lower bounds on the Lyapunov exponent inside the spectral gaps which increase linearly with p (showing that the Lyapunov exponent is large inside the gaps); see Theorem 6.8. By contrast, the Combes–Thomas estimates on the Lyapunov exponent only give the lower bound \(\frac{C}{p}\) (since there is always spectrum at that distance nearby and CT bounds deteriorate close the spectrum), so they significantly underestimate the size of the Lyapunov exponent. See Fig. 1 for a summary.
We remark that a tighter upper bound in the form of a convex optimization problem that can be evaluated efficiently for any fixed a and b is presented in [44].
We mention that there is some degree of flexibility in the choice of \(r_0\), since one can use alternative versions of the effective Avalanche Principle, e.g., Theorem 2.6.
Alternatively it is also possible to use resolvent calculus to relate \(\left\Vert [A_j^{\mathrm {i}t},A_k]\right\Vert \) with \(\left\Vert [A_j,A_k]\right\Vert \).
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Acknowledgements
We thank Jürg Fröhlich, Silvius Klein, Jeffrey Schenker, and Wilhelm Schlag for helpful remarks. DS acknowledges support from the Swiss National Science Foundation via the NCCR QSIT as well as project No. 200020_165843.
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Appendices
Proof of the Avalanche principle, version 2
Proof of Theorem 2.6
The proof follows the general steps in [25]. We only summarize the necessary changes in the argument, which amount to various refined estimates and a different choice of the scaling parameter t (see below).
We define \(\varepsilon _0=\frac{1}{5}\) and \(c_0=\frac{1}{6}\). Using that \((\varepsilon _0+\sqrt{1-\varepsilon _0^2})\pi /2\le 1.86\), the estimate \(\frac{\sqrt{2}\pi }{2}\kappa \varepsilon ^{-2}\) on the Lipschitz constant in Corollary 5.7(b) in [25] can be replaced by
This replacement then propagates through the argument in [25], specifically to Lemma 5.14, Corollary 5.15 and the proof of Theorem 5.5.
Another change is made starting with Lemma 5.12 in [25]. We define \({\tilde{\varepsilon }}=t\varepsilon \) with \(t=\frac{72}{100}\). (This choice of t is close to the largest one allowed here which is important for having a good bound in (A.4) below.) Lemma 5.12 in [25] then holds true with this choice of t under our present assumptions on \(\kappa \) and \(\varepsilon \). To see this, we note that condition (5.7) in [25] can be rearranged as
which is comfortably satisfied for all \(\varepsilon \in (0,\varepsilon _0)\). To establish Lemma 5.12, it then remains to check that
for \(f(x,y)=x\sqrt{1-y^2}+y\sqrt{1-x^2}\), which is tedious, but elementary. (Here we avoid estimating the square roots by their linearization as was done in (5.8) of [25].)
The previously established estimates yield the following modification of Lemma 5.16 in [25]. The relevant Lipschitz constant is now
where the fact that this is strictly less than 1 is important for having a contraction. The relevant estimate after summing the geometric series in the proof of Lemma 5.16 then yields
and then the right-hand side replaces the upper bound from Lemma 5.16 (which was \(3\kappa /\varepsilon \) in [25]).
For the proof of Theorem 5.5(i), it suffices to note that
In the conclusion of Theorem 5.5(i), the bound \(3\frac{\kappa }{\varepsilon }\) is replaced by \(\frac{7}{2}\frac{\kappa }{\varepsilon }\) due to our alternative version of Lemma 5.16 described above.
Finally, in the proof of Theorem 5.5(ii), we replace every instance of \(3\frac{\kappa }{\varepsilon }\) by \(\frac{7}{2}\frac{\kappa }{\varepsilon }\) in (5.17)–(5.19). This has the effect of replacing the lower bound in (5.19) by
and therefore the upper bound in (5.20) by \(\frac{21}{2}\frac{\kappa }{\varepsilon ^2}\). Combining this with the bound (5.21) in [25] and using \(\kappa \le c_0\varepsilon _0^2=\frac{1}{150}\) yields the lower bound on the main quantity
so \(c_l=11\) as desired.
It remains to prove the upper bound on the main quantity. We first note that the upper bound \(\kappa '\) in Lemma 5.17 needs to be replaced by \(\kappa _j'=\left( \frac{3}{5}\right) ^{j-1} \frac{6}{5}\kappa \). Indeed, Lemma 5.16 (with the constant \(\frac{7}{2}\frac{\kappa }{\varepsilon }\le \frac{7}{12}\varepsilon =:r\)) and Proposition 2.29 in [13] imply \(\Vert (D\hat{g_0})_{\hat{\mathfrak {v}}(g^j)}\Vert \le \kappa \frac{r+\sqrt{1-r^2}}{1-r^2}\le \frac{60}{53}\kappa \). From there, applying the chain rule as in [25] implies
with the Lipschitz constant \(L=1.86*\kappa {\tilde{\varepsilon }}^{-2}<\frac{3}{5}\) defined in (A.4). Equipped with this new version of Lemma 5.17 and recalling (A.7), the relevant bound becomes
Since the new upper bound in (5.20) is \(\frac{21}{2}\frac{\kappa }{\varepsilon ^2}\) and we assume \(n\ge 36\), we have
and hence the upper bound on the main quantity
as desired. This proves Theorem 2.6. \(\square \)
Proofs for the almost-commuting case
1.1 Proof of Proposition 5.2
Note that for each index \(1\le k\le n\) it holds that \(A_k^{1+\mathrm {i}t}=A_k A_k^{\mathrm {i}t}=A_k^{\mathrm {i}t} A_k\). We start by rewriting the difference as a telescopic sum
We express the long commutator \([A_j^{\mathrm {i}t},A_{j+1}\ldots A_n]\) as a sum of individual commutators \([A_j^{\mathrm {i}t},A_k ]\) by iteratively applying the Leibniz rule for commutators,
We find
The unitary invariance of the operator norm then gives for \(\varepsilon _t:=\max _{j,k \in \mathbb {N}} \Vert [A_j^{\mathrm {i}t},A_k]\Vert \)
Thus we obtain
where the final step is valid because we can assume without loss of generality that \( 1 \le \varepsilon _t \sum _{j<k \le n} \frac{X_{1,j-1}(t) X_{j,j}(0) X_{j+1,k-1}(0) X_{k+1,n}(0)}{X_{1,n}(0)} \), as otherwise the assertion of Proposition 5.2 holds trivially.
Assumption 5.1 then gives
We thus find
We next bounding the sum with an integral, i.e.,
which is correct because the function \(\phi _{n,c}: [0,n] \ni x \mapsto (n-x)\mathrm {e}^{x c}\) for \(c>0\) is monotonically increasing in \([0,x^\star ]\) and monotonically decreasing in \([x^\star ,n]\) where \(x^\star = n-1/c\). Furthermore we have \(\phi _{n,c}(x^\star )=\frac{1}{c}\mathrm {e}^{nc -1} \le \frac{1}{c}\mathrm {e}^{nc}\). Hence we find
Together with Assumption 5.1 this implies
As a result we obtain
Since this is true for all \(n\in \mathbb {N}\) we can conclude that
which proves the assertion. \(\square \)
1.2 Proof of Corollary 5.3
The n-matrix Golden–Thompson inequality from Theorem 2.1 implies that
where in the first step we swap the limit and the integral which is valid by the dominated convergence theorem. Proposition 5.2 implies
We can use a well known bound [42, Lemma 3.8] to further simplify the boundFootnote 4 by using
which then proves the assertion. \(\square \)
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Lemm, M., Sutter, D. Quantitative lower bounds on the Lyapunov exponent from multivariate matrix inequalities. Anal.Math.Phys. 12, 35 (2022). https://doi.org/10.1007/s13324-021-00641-x
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DOI: https://doi.org/10.1007/s13324-021-00641-x