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One-parameter counterexamples to the refined Bessis-Moussa-Villani conjecture

Hyunho Cha

Abstract

The Bessis-Moussa-Villani (BMV) conjecture, originating in quantum statistical mechanics, was proved by Stahl after an influential reformulation by Lieb and Seiringer. A later refinement asks whether the normalized average over all words with $n$ letters $A$ and $m$ letters $B$ is always bounded above by $\mathrm{tr}(A^nB^m)$ and below by $\mathrm{tr}\exp(n\log A+m\log B)$. We study a specific one-parameter family $(A_x, B_x)$ and derive exact closed formulas for both sides of the first inequality when $n=m=5$. In particular, $x=10^{-3}$ gives a counterexample and, remarkably, the ratio of the normalized word average to the trace $\mathrm{tr}(A^nB^m)$ can become arbitrarily large.

One-parameter counterexamples to the refined Bessis-Moussa-Villani conjecture

Abstract

The Bessis-Moussa-Villani (BMV) conjecture, originating in quantum statistical mechanics, was proved by Stahl after an influential reformulation by Lieb and Seiringer. A later refinement asks whether the normalized average over all words with letters and letters is always bounded above by and below by . We study a specific one-parameter family and derive exact closed formulas for both sides of the first inequality when . In particular, gives a counterexample and, remarkably, the ratio of the normalized word average to the trace can become arbitrarily large.
Paper Structure (6 sections, 1 theorem, 41 equations)

This paper contains 6 sections, 1 theorem, 41 equations.

Table of Contents

  1. Introduction
  2. The one-parameter family and the main formula
  3. Derivation of Eq. \ref{['eq:Lformula']}
  4. Derivation of Eq. \ref{['eq:Rformula']}
  5. Discussion
  6. Python verification script

Key Result

Theorem 1

For every $x\ge 0$, the matrices $A_x$ and $B_x$ are positive semidefinite and Consequently, In particular, $L(10^{-3})<R(10^{-3})$, so the left-hand inequality in Eq. eq:refinement fails for $(A_{10^{-3}},B_{10^{-3}})$.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • proof