Renormalization fixed point of the KPZ universality class

Abstract

The one dimensional Kardar-Parisi-Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a natural renormalization/rescaling on the space of such evolving interfaces. We introduce and describe the renormalization fixed point of the Kardar-Parisi-Zhang universality class in terms of a random nonlinear semigroup with stationary independent increments, and via a variational formula. Furthermore, we compute a plausible formula the exact transition probabilities using replica Bethe ansatz. The semigroup is constructed from the Airy sheet, a four parameter space-time field which is the Airy2 process in each of its two spatial coordinates. Minimizing paths through this field describe the renormalization group fixed point of directed polymers in a random potential. At present, the results we provide do not have mathematically rigorous proofs, and they should at most be considered proposals.

19 pages, 2 figures; updated and expanded version with addition author (Remenik)

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Cited by in corpus (39)

• Growing interfaces uncover universal fluctuations behind scale invariance
• The KPZ fixed point
• The KPZ equation with flat initial condition and the directed polymer with one free end
• The directed landscape
• KPZ equation limit of higher-spin exclusion processes
• Airy processes and variational problems
• Quantum Loewner Evolution
• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
• Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
• From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics
• Shift-invariance for vertex models and polymers
• The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions
• Two time distribution in Brownian directed percolation
• Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class
• Reflected Brownian motions in the KPZ universality class
• Half-space stationary Kardar-Parisi-Zhang equation
• One-sided reflected Brownian motions and the KPZ fixed point
• On global solutions of the random Hamilton-Jacobi equations and the KPZ problem
• How flat is flat in random interface growth?
• Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
• Tail of the two-time height distribution for KPZ growth in one dimension
• Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Entanglement entropy growth in stochastic conformal field theory and the KPZ class
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval
• Riemann surface for TASEP with periodic boundaries
• Crossover between various initial conditions in KPZ growth: flat to stationary
• Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
• Diffusivity of lattice gases
• KPZ fluctuations in finite volume
• TASEP fluctuations with soft-shock initial data
• Fluctuations of TASEP and LPP with general initial data
• Steady state of the KPZ equation on an interval and Liouville quantum mechanics
• Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
• Analytical Methods and Field Theory for Disordered Systems
• Geodesic Forests in the Last-Passage Percolation
• On the local fluctuations of last-passage percolation models