[Paper Review] Large time asymptotics of growth models on space-like paths I: PushASEP

Alexei Borodin, Patrik L. Ferrari|ArXiv.org|Jul 18, 2007

Random Matrices and Applications21 references4 citations

TL;DR

This paper establishes the large-time asymptotic fluctuations of the PushASEP model along arbitrary space-like paths in space-time, proving that these fluctuations converge to the Airy_1 and Airy_2 processes for flat and step initial conditions, respectively. Using determinantal point process techniques and asymptotic analysis of Fredholm determinants, it extends KPZ universality to a new class of asymmetric exclusion processes with pushing dynamics, resolving joint distributions of particle positions across time and space with 1/3 and 2/3 scaling exponents.

ABSTRACT

We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any space-like path are described by the Airy_1 and Airy_2 processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases.

Motivation & Objective

• To establish the large-time fluctuation behavior of the PushASEP model along general space-like paths in space-time.
• To extend KPZ universality to a new interacting particle system that interpolates between TASEP and Toom’s model via a pushing mechanism.
• To analyze joint distributions of particle positions at different times and locations, including tagged particle trajectories and fixed-time height profiles.
• To derive a determinantal formula for the PushASEP with time- and particle-dependent jump rates, enabling rigorous asymptotic analysis.
• To prove convergence of finite-particle systems to infinite-particle limits under flat and step initial conditions, ensuring well-defined dynamics in the large-N limit.

Proposed method

• Derive a determinantal formula for the PushASEP with arbitrary initial conditions using Bethe Ansatz and generalized RSK correspondence techniques.
• Represent the particle position distribution as a gap probability for a signed determinantal point process, enabling Fredholm determinant representation.
• Apply asymptotic analysis of Fredholm determinants via steepest descent methods on the kernel integral, focusing on critical points and scaling limits.
• Rescale space-time coordinates using T^{1/3} and T^{2/3} scaling to extract universal limiting behavior.
• Conjugate the kernel to eliminate linear and quadratic terms in the exponent, isolating the cubic term responsible for the Airy process limit.
• Use dominated convergence and bounds on error terms to justify interchange of limit and summation/integration in the Fredholm series.

Experimental results

Research questions

• RQ1 What are the universal large-time fluctuations of the PushASEP model along arbitrary space-like paths in space-time?
• RQ2 How do the joint distributions of particle positions at different times and locations behave in the large-N, long-time limit?
• RQ3 Does the PushASEP model exhibit KPZ universality with 1/3 and 2/6 scaling exponents, even in the absence of drift?
• RQ4 Can the infinite-particle PushASEP be rigorously defined as a limit of finite systems under flat and step initial conditions?
• RQ5 What is the limiting process for the height function of the associated growth model along space-like paths?

Key findings

• For flat initial conditions, the large-time fluctuations of the height function along any space-like path converge to the Airy_1 process.
• For step initial conditions, the large-time fluctuations converge to the Airy_2 process, confirming KPZ universality in a new model.
• The fluctuation exponent is 1/3 even when the drift is zero, due to the intrinsic asymmetry in the pushing dynamics.
• The joint distribution of particle positions at different times and locations converges to a Fredholm determinant with the extended Airy kernel.
• The asymptotic analysis confirms that the limiting kernel matches the extended Airy kernel associated with the Airy_2 process after appropriate scaling.
• The convergence is uniform and dominated, allowing rigorous interchange of limits via the dominated convergence theorem.