Metadata
bibkey: ayyermallick2010annihilation
authors: A. Ayyer and K. Mallick
year: 2010
title: "Exact results for an asymmetric annihilation process with open boundaries"
doi: 10.1088/1751-8113/43/4/045003
url: https://arxiv.org/abs/0910.0693
claim: "After Eq. (79): each eigenvalue seems to have exactly one eigenvector independent of the number of times the eigenvalue appears."
strata_touched:
- D5/S3/StatisticalMechanics/ParticleProcesses/AnnihilationMaximalDegeneracyRefutation
license: citation-only
triage: anchor
Verified locator
DOI: https://doi.org/10.1088/1751-8113/43/4/045003
Source: https://arxiv.org/abs/0910.0693
Model and conjecture
A. Ayyer and K. Mallick, Exact results for an asymmetric annihilation process with open boundaries, J. Phys. A 43, 045003 (2010). Page numbers for this source refer to arXiv:0910.0693v2.
Section II, p. 2:
We consider a nonequilibrium system on a finite lattice with L sites labelled from 1 to L.
Empty sites are denoted by 0.
The displayed rules (1)–(3), p. 2, are:
10 → 01 with rate 1, 11 → 00 with rate λ.
0 → 1 with rate α, 1 → 0 with rate αλ.
1 → 0 with rate β.
The specialization λ = 1 makes the two bulk rates equal and makes the left boundary a symmetric flip. A Boolean function on Fin L represents a state, with true for a particle and index 0 for site 1. Bulk rates are added over all bonds; left and right boundary rates are added when their moves have the specified target state.
After Eq. (79), p. 14:
We also conjecture that the Markov matrices are maximally undiagonalizable in the sense that each eigenvalue seems to have exactly one eigenvector independent of the number of times the eigenvalue appears (this implies that the Jordan blocks are of the maximum possible size). In other words, the degeneracy seems to be so strong that the Markov matrix M_L which is of size 2^L has only 2L eigenvectors.
The formal claim uses geometric multiplicity at most one for every real eigenvalue, every positive length, and all positive real α and β. Restricting to real eigenvalues is sufficient for the refutation at −2: at L = 5, α = 1/3 and β = 2/5, the eigenspace has dimension at least two. The claim does not assert existence of an eigenvector for a number outside the spectrum.
Ayyer–Strehl restatement
A. Ayyer and V. Strehl, The spectrum of an asymmetric annihilation process, FPSAC 2010, DMTCS Proc. AN, 461–472; DOI 10.46298/dmtcs.2884; https://arxiv.org/abs/1008.2134v1.
Definition 1, p. 463 (PDF p. 3):
A (continuous-time) transition matrix or Markov matrix or stochastic matrix is a square matrix of size equal to the cardinality of the configuration space whose (i, j)-th entry is given by the rate of the transition from configuration j to configuration i, when i is not equal to j. The (i, i)-th entry is then fixed by demanding that the entries in each column sum to zero.
Conjecture 1, p. 466 (PDF p. 6):
The matrix M_L is maximally degenerate in the sense that it has exactly 2L eigenvectors.
Section 5, p. 471 (PDF p. 11):
Our proof does not seem to explain the maximum amount of degeneracy, as stated in Conjecture 1.
Theorem 4 describes the characteristic polynomial. A geometric-multiplicity counterexample does not contradict that characteristic-polynomial result.