General position of a projection and its image under a free unitary Brownian motion

TL;DR

This paper proves that a projection and its conjugation by a free unitary Brownian motion are in general position at all times, using PDEs and spectral flow analysis in free probability theory.

Contribution

It introduces a PDE-based method to analyze the spectral distribution dynamics of projections under free unitary Brownian motion, establishing general position at all times.

Findings

Proves projections and their conjugates are in general position at any time.

Derives a PDE for the spectral distribution's Herglotz transform.

Provides a flow on [-1,1] linking initial and stationary spectral distributions.

Abstract

Given an orthogonal projection $P$ and a free unitary Brownian motion $Y = (Y_t)_{t \geq 0}$ in a $W^{\star}$-non commutative probability space such that $Y$ and $P$ are $\star$-free in Voiculescu's sense, the main result of this paper states that $P$ and $Y_tPY_t^{\star}$ are in general position at any time $t$. To this end, we study the dynamics of the unitary operator $SY_tSY_t^{\star}$ where $S = 2P-1$. More precisely, we derive a partial differential equation for the Herglotz transform of its spectral distribution, say $\mu_t$. Then, we provide a flow on the interval $[-1,1]$ in such a way that the Herglotz transform of $\mu_t$ composed with this flow is governed by both the Herglotz transforms of the initial ($t=0$) and the stationary ($t = \infty)$ distributions. This fact allows to compute the weight that $\mu_t$ assigns to $z=1$ leading to the main result. As a by-product, the weight that the spectral distribution of the free Jacobi process assigns to $x=1$ follows after a normalization. In the last part of the paper, we use combinatorics of non crossing partitions in order to analyze the term corresponding to the exponential decay $e^{-nt}$ in the expansion of the $n$-th moment of $SY_tSY_t^{\star}$.