2023 Theses Doctoral

# Path properties of KPZ models

In this thesis we investigate large deviation and path properties of a few models within the Kardar-Parisi-Zhang (KPZ) universality class.

The KPZ equation is the central object in the KPZ universality class. It is a stochastic PDE describing various objects in statistical mechanics such as random interface growth, directed polymers, interacting particle systems. In the first project we study one point upper tail large deviations of the KPZ equation ๐ข(t,x) started from narrow wedge initial data. We obtain precise expression of the upper tail LDP in the long time regime for the KPZ equation. We then extend our techniques and methods to obtain upper tail LDP for the asymmetric exclusion process model, which is a prelimit of the KPZ equation.

In the next direction, we investigate temporal path properties of the KPZ equation. We show that the upper and lower law of iterated logarithms for the rescaled KPZ temporal process occurs at a scale (log log ๐ก)ยฒ/ยณ and (log log ๐ก)ยน/ยณ respectively. We also compute the exact Hausdorff dimension of the upper level sets of the solution, i.e., the set of times when the rescaled solution exceeds ๐ผ(log log ๐ก)ยฒ/ยณ. This has relevance from the point of view of fractal geometry of the KPZ equation.

We next study superdiffusivity and localization features of the (1+1)-dimensional continuum directed random polymer whose free energy is given by the KPZ equation. We show that for a point-to-point polymer of length ๐ก and any ๐ โฒ (0,1), the point on the path which is ๐๐ก distance away from the origin stays within a ๐(1) stochastic window around a random point ๐_๐,๐ก that depends on the environment. This provides an affirmative case of the folklore `favorite region' conjecture. Furthermore, the quenched density of the point when centered around ๐_๐,๐ก converges in law to an explicit random density function as ๐ก โ โ without any scaling. The limiting random density is proportional to ๐^{-๐ก(x)} where ๐ก(x) is a two-sided 3D Bessel process with diffusion coefficient 2. Our proof techniques also allow us to prove properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. In a follow up project, we show that the annealed law of polymer of length ๐ก, upon ๐กยฒ/ยณ superdiffusive scaling, is tight (as ๐ก โ โ) in the space of ๐ถ([0,1]) valued random variables. On the other hand, as ๐ก โ 0, under diffusive scaling, we show that the annealed law of the polymer converges to Brownian bridge.

In the final part of this thesis, we focus on an integrable discrete half-space variant of the CDRP, called half-space log-gamma polymer.We consider the point-to-point log-gamma polymer of length 2๐ in a half-space with i.i.d.Gammaโปยน(2๐ณ) distributed bulk weights and i.i.d. Gammaโปยน(๐ผ+๐ณ) distributed boundary weights for ๐ณ > 0 and ๐ผ > -๐ณ. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when ๐ผ โฅ 0. In particular, in this regime, we show that after appropriate centering, the free energy process with spatial coordinate scaled by ๐ยฒ/ยณ and fluctuations scaled by ๐ยน/ยณ is tight.

The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles. The ๐ผ โฅ 0 regime correspond to a polymer measure which is not pinned at the boundary. In a companion work, we investigate the ๐ผ < 0 setting. We show that in this case, the endpoint of the point-to-line polymer stays within ๐(1) window of the diagonal. We also show that the limiting quenched endpoint distribution of the polymer around the diagonal is given by a random probability mass function proportional to the exponential of a random walk with log-gamma type increments.

## Subjects

## Files

- Das_columbia_0054D_17727.pdf application/pdf 4.38 MB Download File

## More About This Work

- Academic Units
- Mathematics
- Thesis Advisors
- Corwin, Ivan Z.
- Degree
- Ph.D., Columbia University
- Published Here
- April 12, 2023