Critical Gaussian Multiplicative Chaos revisited

TL;DR

This paper provides new, concise proofs demonstrating the convergence of four different sequences to the critical Gaussian Multiplicative Chaos, enhancing understanding of this complex stochastic process.

Contribution

It introduces simplified, self-contained proofs for the convergence of multiple sequences to the critical Gaussian Multiplicative Chaos, clarifying previous results.

Findings

Proves convergence of the derivative martingale to critical GMC

Establishes convergence of the critical martingale and mollified field exponential

Demonstrates convergence of subcritical GMC sequences

Abstract

We present new, short and self-contained proofs of the convergence (with an adequate renormalization) of four different sequences to the critical Gaussian Multiplicative Chaos:(a) the derivative martingale (b) the critical martingale (c) the exponential of the mollified field (d) the subcritical Gaussian Multiplicative Chaos.