A Riemann‐Hilbert Approach to the Lower Tail of the Kardar‐Parisi‐Zhang Equation

Abstract

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar‐Parisi‐Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distributions in models of positive‐temperature free fermions. We show that logarithmic derivatives of the Fredholm determinants can be expressed in terms of a 2 × 2 Riemann‐Hilbert problem, and we use this to derive asymptotics for the Fredholm determinants. As an application of our result, we derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal. © 2021 Wiley Periodicals LLC.

Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation

“…Other proofs and extensions of this integro-differential equation have also been recently found in related contexts [22,23,46]. Also, by exploring (2.12) the tail behavior of the KPZ equation has become rigorously accessible in various asymptotic regimes [24,25,27,30]. As a first result, we prove that the multiplicative statistics L Ai is the universal limit of L Q n (s).…”

“…In our case, a novel feat is that this local parametrix construction is performed in a two-step way, first with the construction of a model problem varying with large parameter, and second with the asymptotic analysis of this model problem. In the latter, a RHP recently studied by Cafasso and Claeys [24] (see also the subsequent works [25,27]) which is related to the lower tail of the KPZ equation shows up, and it is this RHP that ultimately connects all of our considered quantities to the integro-differential PII.…”

“…One of the key aspects of this representation is that the Airy 2 point process is determinantal, and consequently its statistics can be studied using techniques from exactly solvable/integrable models. Indeed Equation 2.12 is the starting point taken by Cafasso and Claeys [24], who then connected L Ai (s, T ) to a RHP that will also play a major role for us. Recently, Cafasso, Claeys and Ruzza [25] also obtained an independent proof of the representation (2.10), extending it to more general multiplicative statistics of the Airy 2 point process.…”

“…The expression (2.5) may be viewed as a transformation of the point process, where ζ ∈ C becomes the spectral variable of this transformation, and the matching ζ = e −s motivates the distinguished role of s in (2.3). In the context of random particle systems, this particular multiplicative statistics is associated to the notion of a q-Laplace transform [16][17][18][19] that we already mentioned in the Introduction, and it has been one of the key quantities in several outstanding recent progresses in asymptotics for random particle systems [24,30,43]. We work under the following assumptions.…”

“…This program has already been taken to a great start for the KPZ equation, and exploring its connection with the Airy 2 point process Corwin and Ghosal [30] were able to obtain bounds for the lower tail of the KPZ equation. Shortly afterwards, such bounds were improved by Cafasso and Claeys [24] with Riemann-Hilbert methods common in random matrix theory.…”