Probabilistic Perspective Toward KPZ Class Models

LINGFU ZHANG- UNIVERSITY OF CALIFORNIA, BERKELEY 

ABSTRACT

A striking phenomenon in probability theory is universality, where different probabilistic models produce the same large-scale or long-time limit. One example is the Kardar-Parisi-Zhang (KPZ) universality class, encompassing a wide range of natural models such as growth processes modeling bacterial colonies, eigenvalues of random matrices, and traffic flow models originating from mRNA translation. Historically, these KPZ class models have mostly been studied via algebraic methods. In this talk, I will introduce a general strategy that adopts a more probabilistic perspective. This approach has enabled us to successfully resolve many open problems. I will present a selection of these results, including (1) local statistics of Last Passage Percolation – a pivotal model in the KPZ class, (2) the slow bond problem in a related traffic flow model, and (3) the cutoff phenomenon in Metropolis biased card shuffling. No prior knowledge of the topic will be assumed.

Related Papers: 

• https://arxiv.org/abs/2106.05242
• https://arxiv.org/abs/2109.04563
• https://arxiv.org/abs/2208.13383