Physics

Exponential function

Operator (biology)

Probability density function

Mathematics

Mathematical physics

Transcription factor

Moment-generating function

Gene

Statistics

Mathematical analysis

Airy function

Repressor

Rate function

Moment (physics)

Chemistry

Geometry

Differential equation

Applied mathematics

Riccati equation

Wedge (geometry)

Biochemistry

Classical mechanics

Large deviations theory

Scaling

Duke Mathematical Journal

Consider the Hopf–Cole solution h(t,x) of the Kandar–Parisi–Zhang (KPZ) equation with the narrow wedge initial condition. Regarding t→∞ as a scaling parameter, we provide the first rigorous proof of the large deviation principle (LDP) for the lower tail of h(2t,0)+t 12, with speed t2 and an explicit rate function Φ −(z). This result confirms existing physics predictions made by Corwin (2011); Sasorov, Meerson, and Prolhac (2017); and Krajenbrink, Le Doussal, and Prolhac (2018). Our analysis utilizes a formula from Borodin and Gorin (2016) to convert the LDP for the KPZ equation to calculating an exponential moment of the Airy point process (PP). To estimate this exponential moment, we invoke the stochastic Airy operator (SAO) and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

Exact lower-tail large deviations of the KPZ equation