arXiv subjects

Juniper Bahr

Publications and source records attributed to Juniper Bahr.

2 recordsLinked to original sources

arXiv2021

Free Moment Measures and Laws

In arXiv:1304.0630, it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on $\mathbb{R}^n$ by the map $U \mapsto (\nabla U)_{\#} e^{-U} dx$. We consider whether there is a similar coordinatization of non-commutative probability spaces, with the Gibbs measure $e^{-U} dx$ replaced by the corresponding free Gibbs law. We call laws parametrized in this way free moment laws. We first consider the case of a single (and thus commutative) random variable and then the regime of $n$ non-commutative random variables which are perturbations of freely independent semi-circular variables. We prove that free moment laws exist with little restriction for the one dimensional case, and for small even perturbations of free semi-circle laws in the general case.

math.OA↗

arXiv2016

Subrings of $\mathbb{C}$ Generated by Angles

Consider the following inductively defined set. Given a collection $U$ of unit magnitude complex numbers, and a set initially containing just 0 and 1, through each point in the set, draw lines whose angles with the real axis are in $U$. Add every intersection of such lines to the set. Upon taking the closure, we obtain $R(U)$. We investigated for which $U$, $R(U)$ is a ring. Our main result holds for when $1 \in U$ and $|U| \ge 4$. If $P$ is the set of real numbers in $R(U)$ generated in the second step of the construction, then $R(U)$ equals the module over $\mathbb{Z}[P]$ generated by the set of points made in the first step of the construction. This lets us show that whenever the pairwise products of points made in the first step remain inside $R(U)$, it is closed under multiplication, and is thus a ring.

math.RA↗