Further Implications of the Bessis–Moussa–Villani Conjecture

Abstract

We find further implications of the BMV conjecture, which states that for hermitian matrices B≥0 and A, the function $\lambda\mapsto \operatorname {Tr}\exp(\mathsf {A}-{\lambda }\mathsf {B})$ is the Laplace transform of a positive measure supported on [0,∞].

Appendix: Proof of Lemma 1 for \(p\in\pmb{\mathbb {N}}\)

By induction it is easy to show that

(5)

By taking the trace at λ=0 we obtain

(6)

Moreover, by similar arguments,

(7)

By taking the trace at λ=0 and using cyclicity, we get

(8)

We have to show that

$$ I_2=\frac{p}{p+r} (-1)^r I_1\ . $$

(9)

To see this we rewrite I ₁ in the following way. Define p+r matrices M _j by

$$ \mathsf {M}_j= \left\{\begin{array}{@{}l@{\quad }l}\mathsf {B}& \mbox{for\ }1\leq j\leq r \\[3pt] \mathsf {A}& \mbox{for\ } r+1\leq j\leq r+p . \end{array}\right. $$

(10)

Let $\mathcal {S}_{n}$ denote the permutation group. Then

$$ I_1=\frac{1}{p!} \sum_{\pi\in \mathcal {S}_{p+r}} \operatorname {Tr}\prod _{j=1}^{p+r} \mathsf {M}_{\pi(j)} . $$

(11)

Because of the cyclicity of the trace we can always arrange the product such that M _p+r has the first position in the trace. Since there are p+r possible locations for M _p+r to appear in the product above, and all products are equally weighted, we get

$$ I_1=\frac {p+r}{p!} \sum_{\pi\in \mathcal {S}_{p+r-1}} \operatorname {Tr}\mathsf {A}\prod_{j=1}^{p+r-1} \mathsf {M}_{\pi(j)} . $$

(12)

On the other hand,

$$ I_2=(-1)^r \frac{1}{(p-1)!} \sum _{\pi\in \mathcal {S}_{p+r-1}} \operatorname {Tr}\mathsf {A}\prod_{j=1}^{p+r-1} \mathsf {M}_{\pi(j)} , $$

(13)

so we arrive at the desired equality.