Paving property for real stable polynomials and strongly Rayleigh processes (Q2137014)

One important aspect of the Kadison-Singer problem is that it has been shown to be equivalent to a large number of problems in various fields. For a background on the Kadison-Singer problem and its equivalent formulations see [\textit{M. Bownik}, Contemp. Math. 706, 63--92 (2018; Zbl 1401.42032)]. One of the equivalent formulations, which is the main focus of the present paper, is as follows. Problem. Let \(\varepsilon \in (0, 1)\). Does there exist \(r\in\mathbf{N}\) such that every Hermitian matrix \(A\) whose diagonal entries are zero can be \((r, \varepsilon)\)-paved, namely there are diagonal projections \(P_1, \dots, P_r\) such that \(\sum_{i=1}^r P_i = I\) and for any \(i\in [r]\) the following inequality \(\|P_iA P_i\|_{op} \le \| A\|_{op}\) is true, where \([r] = \{1,\dots, r\}\) and \(\|\cdot\|_{op}\) denotes the operator norm. This formulation of the Kadison-Singer problem is known as the paving conjecture. The authors first extend the result in [\textit{A. W. Marcus} et al., Ann. Math. (2) 182, No. 1, 327--350 (2015; Zbl 1332.46056)] to real stable polynomials. The authors prove that for every multi-affine real stable polynomial satisfying a simple condition, it is possible to partition its set of variables to a small number of subsets such that the ``restriction'' of the polynomial to each subset has small roots, see also [\textit{M. Ravichandran} and \textit{J. Leake}, Math. Ann. 377, No. 1--2, 511--541 (2020; Zbl 1482.46074)].