PreprintCC BY 4.0arXiv
Double-Scaling Limits for Monitored Haar Products
Abstract
We study products of independent Haar matrices interspersed with a fixed corank-one projection. As the matrix dimension N and product length L grow with L/N→τ, we prove that the empirical distribution of squared singular values converges to the free multiplicative Poisson law with S-transform S(z)=exp(τ/(1+z)), for both the unitary and orthogonal ensembles. Lagrange inversion yields its moments explicitly, proving Beenakker's conjectured Erlang formula for every moment order.
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BibTeX
@misc{lee2026doublescaling,
title = {{Double-Scaling Limits for Monitored Haar Products}},
author = {Joon Hyung Lee},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.05693},
eprint = {2607.05693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.05693},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Double-Scaling Limits for Monitored Haar Products
AU - Joon Hyung Lee
PY - 2026
DA - 2026/11/09
PB - arXiv
DO - 10.48550/arxiv.2607.05693
UR - https://arxiv.org/abs/2607.05693
AB - We study products of independent Haar matrices interspersed with a fixed corank-one projection. As the matrix dimension N and product length L grow with L/N→τ, we prove that the empirical distribution of squared singular values converges to the free multiplicative Poisson law with S-transform S(z)=exp(τ/(1+z)), for both the unitary and orthogonal ensembles. Lagrange inversion yields its moments explicitly, proving Beenakker's conjectured Erlang formula for every moment order.
ER -