تقرير
Liouville quantum gravity from random matrix dynamics
| العنوان: | Liouville quantum gravity from random matrix dynamics |
|---|---|
| المؤلفون: | Bourgade, Paul, Falconet, Hugo |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematics - Probability, Mathematical Physics |
| الوصف: | We establish the first connection between $2d$ Liouville quantum gravity and natural dynamics of random matrices. In particular, we show that if $(U_t)$ is a Brownian motion on the unitary group at equilibrium, then the measures $$ |\det(U_t - e^{i \theta})|^{\gamma} dt d\theta $$ converge in the limit of large dimension to the $2d$ LQG measure, a properly normalized exponential of the $2d$ Gaussian free field. Gaussian free field type fluctuations associated with these dynamics were first established by Spohn (1998) and convergence to the LQG measure in $2d$ settings was conjectured since the work of Webb (2014), who proved the convergence of related one dimensional measures by using inputs from Riemann-Hilbert theory. The convergence follows from the first multi-time extension of the result by Widom (1973) on Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols. To prove these, we develop a general surgery argument and combine determinantal point processes estimates with stochastic analysis on Lie group, providing in passing a probabilistic proof of Webb's $1d$ result. We believe the techniques will be more broadly applicable to matrix dynamics out of equilibrium, joint moments of determinants for classes of correlated random matrices, and the characteristic polynomial of non-Hermitian random matrices. Comment: v3: fixes a minor error in the proof of Proposition 3.4, 40 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2206.03029 |
| رقم الأكسشن: | edsarx.2206.03029 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |