Theses Doctoral

Path properties of KPZ models

Das, Sayan

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 ๐œข(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 ๐‘ก)ยฒ/ยณ and (log log ๐‘ก)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 ๐›ผ(log log ๐‘ก)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 ๐‘ก and any ๐‘ โ‹ฒ (0,1), the point on the path which is ๐‘๐‘ก distance away from the origin stays within a ๐‘‚(1) stochastic window around a random point ๐™ˆ_๐‘,๐‘ก 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™ˆ_๐‘,๐‘ก converges in law to an explicit random density function as ๐‘ก โ†’ โˆž without any scaling. The limiting random density is proportional to ๐‘’^{-๐“ก(x)} where ๐“ก(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 ๐‘ก, upon ๐‘กยฒ/ยณ superdiffusive scaling, is tight (as ๐‘ก โ†’ โˆž) in the space of ๐ถ([0,1]) valued random variables. On the other hand, as ๐‘ก 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.Gammaโปยน(2๐›ณ) distributed bulk weights and i.i.d. Gammaโปยน(๐›ผ+๐›ณ) distributed boundary weights for ๐›ณ > 0 and ๐›ผ > -๐›ณ. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when ๐›ผ โ‰ฅ 0. In particular, in this regime, we show that after appropriate centering, the free energy process with spatial coordinate scaled by ๐‘ยฒ/ยณ and fluctuations scaled by ๐‘ยน/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 ๐›ผ โ‰ฅ 0 regime correspond to a polymer measure which is not pinned at the boundary. In a companion work, we investigate the ๐›ผ < 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.


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More About This Work

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