Free positive multiplicative Brownian motion and the free additive convolution of semicircle and uniform distribution
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2025
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| author | Auer, Martin |
| author_facet | Auer, Martin |
| contents | The free positive multiplicative Brownian motion $(h_t)_{t\geq0}$ is the large $N$ limit in non-commutative distribution of matrix geometric Brownian motion. It can be constructed by setting $h_t:=g_{t/2}g_{t/2}^*$, where $(g_t)_{t\geq0}$ is a free multiplicative Brownian motion, which is the large $N$ limit in non-commutative distribution of the Brownian motion in $\operatorname{Gl}(N,\mathbb{C})$. One key property of $(h_t)_{t\geq0}$ is the fact that the corresponding spectral distributions $(ν_t)_{t\geq0}\subset M^1((0,\infty))$ form a semigroup w.r.t. free multiplicative convolution. In recent work by M. Voit and the present author, it was shown that $ν_t$ can be expressed by the image measure of a free additive convolution of the semicircle and the uniform distribution on an interval under the exponential map. In this paper, we provide a new proof of this result by calculating the moments of the free additive convolution of semicircle and uniform distributions on intervals. As a by-product, we also obtain new integral formulas for $ν_t$ which generalize the corresponding known moment formulas involving Laguerre polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_05984 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free positive multiplicative Brownian motion and the free additive convolution of semicircle and uniform distribution Auer, Martin Probability Combinatorics Functional Analysis Operator Algebras Primary 46L54, Secondary 60B20, 60E10, 05A19, 33C15 The free positive multiplicative Brownian motion $(h_t)_{t\geq0}$ is the large $N$ limit in non-commutative distribution of matrix geometric Brownian motion. It can be constructed by setting $h_t:=g_{t/2}g_{t/2}^*$, where $(g_t)_{t\geq0}$ is a free multiplicative Brownian motion, which is the large $N$ limit in non-commutative distribution of the Brownian motion in $\operatorname{Gl}(N,\mathbb{C})$. One key property of $(h_t)_{t\geq0}$ is the fact that the corresponding spectral distributions $(ν_t)_{t\geq0}\subset M^1((0,\infty))$ form a semigroup w.r.t. free multiplicative convolution. In recent work by M. Voit and the present author, it was shown that $ν_t$ can be expressed by the image measure of a free additive convolution of the semicircle and the uniform distribution on an interval under the exponential map. In this paper, we provide a new proof of this result by calculating the moments of the free additive convolution of semicircle and uniform distributions on intervals. As a by-product, we also obtain new integral formulas for $ν_t$ which generalize the corresponding known moment formulas involving Laguerre polynomials. |
| title | Free positive multiplicative Brownian motion and the free additive convolution of semicircle and uniform distribution |
| topic | Probability Combinatorics Functional Analysis Operator Algebras Primary 46L54, Secondary 60B20, 60E10, 05A19, 33C15 |
| url | https://arxiv.org/abs/2505.05984 |