TL;DR
This paper introduces a new diffusion-based method for calculating averages of ratios and products of characteristic polynomials in Random Matrix Theory, offering exact finite-size formulas and large N asymptotics.
Contribution
It presents a novel diffusion method based on Dyson Brownian motion and Grassmann integration, providing an alternative to existing RMT techniques for Gaussian measures.
Findings
Exact formulas for finite N
Integral representations for large N asymptotics
Explicit calculations demonstrating method applications
Abstract
We introduce a simple yet powerful calculational tool useful in calculating averages of ratios and products of characteristic polynomials. The method is based on Dyson Brownian motion and Grassmann integration formula for determinants. It is intended as an alternative to other RMT techniques applicable to general gaussian measures. Resulting formulas are exact for finite matrix size N and form integral representations convenient for large N asymptotics. Quantities obtained by the method can be interpreted as averages over matrix models with an external source. We provide several explicit and novel calculations showing a range of applications.
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