Presented By: Department of Mathematics
Special Events Seminar
Dissertation Defense: Topics in interacting particle systems and random Schr\'odinger operators
In this dissertation, we study some large probabilistic systems with strong correlations mainly coming from mathematical physics. The dissertation is split into two main parts.
The first part is about the totally asymmetric simple exclusion process (TASEP) which is a default probabilistic model for one-dimensional traffic transport. It also serves as a prototypical example among the so-called Kardar-Parisi-Zhang universality class consisting of a large class of strongly correlated random systems modeling random interface growth.
TASEP has a rich algebraic/combinatorial structure which leads to exact formulas for the joint distributions (the study of such probabilistic systems with rich algebraic structures is known as integrable probability). We will explain how the multi-point space-time joint distributions of TASEP (and some variants) can be derived using techniques from combinatorics (symmetric functions), quantum integrable systems and complex analysis. We will also study the large-scale long-time behaviors of such systems and they exhibit interesting phase transitions when we introduce certain inhomogeneity.
The second part is about random Schr\"odinger operators (RSOs) describing quantum evolutions in disordered media. We will mainly focus on the behaviors of the spectra of RSOs under spatial conditioning (i.e., what can we say if we know some partial information of the spectrum). In particular, a property known as the number rigidity is established for the eigenvalue point processes of a large class of RSOs. Number rigidity roughly states that the total (random) number of eigenvalues inside any bounded set is a deterministic function of the eigenvalue configurations outside of the bounded set. The main tools we use are probabilistic representations of a special class of exponential linear spectral statistics which is related to the semigroups associated with the RSOs and their traces.
Yuchen's advisors are Jinho Baik and Raj Rao Nadakuditi.
Zoom: https://umich.zoom.us/j/91480393821
Passcode: 569413 Speaker(s): Yuchen Liao (UM)
The first part is about the totally asymmetric simple exclusion process (TASEP) which is a default probabilistic model for one-dimensional traffic transport. It also serves as a prototypical example among the so-called Kardar-Parisi-Zhang universality class consisting of a large class of strongly correlated random systems modeling random interface growth.
TASEP has a rich algebraic/combinatorial structure which leads to exact formulas for the joint distributions (the study of such probabilistic systems with rich algebraic structures is known as integrable probability). We will explain how the multi-point space-time joint distributions of TASEP (and some variants) can be derived using techniques from combinatorics (symmetric functions), quantum integrable systems and complex analysis. We will also study the large-scale long-time behaviors of such systems and they exhibit interesting phase transitions when we introduce certain inhomogeneity.
The second part is about random Schr\"odinger operators (RSOs) describing quantum evolutions in disordered media. We will mainly focus on the behaviors of the spectra of RSOs under spatial conditioning (i.e., what can we say if we know some partial information of the spectrum). In particular, a property known as the number rigidity is established for the eigenvalue point processes of a large class of RSOs. Number rigidity roughly states that the total (random) number of eigenvalues inside any bounded set is a deterministic function of the eigenvalue configurations outside of the bounded set. The main tools we use are probabilistic representations of a special class of exponential linear spectral statistics which is related to the semigroups associated with the RSOs and their traces.
Yuchen's advisors are Jinho Baik and Raj Rao Nadakuditi.
Zoom: https://umich.zoom.us/j/91480393821
Passcode: 569413 Speaker(s): Yuchen Liao (UM)
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