The Law of the Sum of the Two Faces of Bi-Monotone Brownian Motion Has Atoms: A Partial Answer to a Question of M. Gerhold

M. Gerhold asked for a concrete description of the vacuum distributions \(\mu_t\) of the operators \(\lambda_t^*+\lambda_t+\rho_t^*+\rho_t\) on the monotone Fock space, that is, of the sum of the two faces of bi-monotone Brownian motion (Oberwolfach Rep. 15 (2018)); the same question closes his paper on bi-monotone Brownian motion, where the even moments of \(\mu_1\) are identified as \(\#\mathrm{PP}_{\bowtie}(2n)/n!\) with the numbers \(1,4,48,928,24448,\dots\) of bi-monotone pair partitions, for which no explicit or recursive formula was known. We give a partial answer: we do not find a closed formula, and the question remains open; we prove structural facts. (1) The measures \(\mu_t\) are the dilations by \(\sqrt t\) of one measure \(\mu=\mu_1\); this scaling is implicit in the source. (2) The compression of the operator to the orthogonal complement of the vacuum has norm at most \(\sqrt6\), while the operator itself has norm \(c>\sqrt6\). Hence the edge \(c\) of the support of \(\mu\) is an isolated atom: \(\mu=w(\delta_c+\delta_{-c})+\mu_0\) with \(\mu_0\) carried by \([-\sqrt6,\sqrt6]\). This is proved without computation. Enclosures in exact rational arithmetic (computer-assisted) give \(c=2.585826606048204448\ldots\) and \(w=0.251331654638419314\ldots\), so the two atoms carry more than half of the mass. (3) The Boolean cumulants of \(\mu\) are \(k_{n+2}=4\langle B^n1\rangle\) for an explicit linear operator \(B\) on polynomials in three variables and an explicit linear functional. This gives all moments exactly in polynomial time; the table of the source (eleven values) is extended to \(m_{306}\). (4) Computer-assisted, exact: \(\mu\) has mass in \((2.01148,2.01149)\) and in \((2.001,2.003)\), so \(\mu_0\) is not carried by \([-2,2]\). Only observed numerically, and not proved: further spectral points accumulating at \(\pm2\) that look like atoms, a nearly flat density of total mass about \(0.49\) on \([-2,2]\), Jacobi parameters tending slowly to \(1\), and the absence of a linear recurrence with polynomial coefficients for the moments within the sizes searched. An explicit formula for \(\mu\), the nature of its part in \([-\sqrt6,\sqrt6]\), and a formula for the numbers \(\#\mathrm{PP}_{\bowtie}(2n)\) remain open. This is an unrefereed note.