This thesis investigates the geometry of polymers and other miscellaneous results in the Kardar-Parisi-Zhang (KPZ) universality class. Directed polymers have enjoyed a rich history in both probability theory and mathematical physics and have connections to several families of statistical mechanical and random growth models that belong to the KPZ universality class [77]. In this thesis, we focus on 2 integrable polymer models, the (1+1)-dimensional continuum directed random polymer (CDRP) and the half-space log-gamma (HSLG) polymer, and study their path properties. For the CDRP, we show both of its superdiffusivity and localization features. Namely, the annealed law of polymer of length t, upon t²/³ superdiffusive scaling, is tight in the space of C( [0, 1])-valued random variables and the quenched law of any point distance pt from the origin on the path a point-to-point polymer (or the endpoint of a point-to-line polymer) concentrates in a O(1) window around a random favorite point Mp,t.
The former marks the first pathwise tightness result for positive temperature models and the latter result confirms the “favorite region conjecture” for the CDRP. Moreover, we provide an explicit random density for the quenched distribution around the favorite point Mp,t. The proofs of both results utilize connections with the KPZ equation and our techniques also allow us to prove properties of the KPZ equation itself, such as ergodicity and limiting Bessel behaviors around the maximum. For the HSLG polymers, we combine our localization techniques from the CDRP and the recently developed HSLG line ensemble results [22, 27] with an innovative combinatorial argument to obtain its limiting quenched endpoint distribution from the diagonal in the boundphase (α < 0).
This result proves Kardar’s “pinning” conjecture in the case of HSLG polymers[158]. Finally, this thesis also contains two separate works on the tightness of the Bernoulli Gibbsian line ensemble under mild conditions and the upper-tail large deviation principle (LDP) of the asymmetric simple exclusion process (ASEP) with step initial data. In the first work, we prove that under a mild but uniform control of the one-point marginals of the top curve of the line ensemble, i.e. the shape of the top curve as approximately an inverse parabola and asymptotically covering the entire real line after scaling and recentering, the sequence of line ensembles is tight.
With a characterization of [109], our tightness result implies the convergence of the Bernoulli Gibbsian line ensemble to the parabolic Airy line ensemble if the top curve converges to the parabolic Airy2 process in the finite dimensional sense. Compared to a similar work of [93], our result applies to line ensembles with possibly random initial and terminal data, instead of a packed initial condition, and does not rely on exact formulas. In our work on the ASEP, we obtain the exact Lyapunov exponent for the height function of ASEP with step initial data and subsequently its upper-tail LDP, where the rate function matches with that of the TASEP given in a variational form in [156].
| Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/n7tj-0623 |
| Date | January 2023 |
| Creators | Zhu, Weitao |
| Source Sets | Columbia University |
| Language | English |
| Detected Language | English |
| Type | Theses |
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