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Asymptotic Properties of Disordered Systems / Jiaming Xia.
- Format:
- Book
- Thesis/Dissertation
- Author/Creator:
- Xia, Jiaming, author.
- Language:
- English
- Subjects (All):
- Mathematics.
- Mathematics--Penn dissertations.
- Penn dissertations--Mathematics.
- Local Subjects:
- Mathematics.
- Mathematics--Penn dissertations.
- Penn dissertations--Mathematics.
- Physical Description:
- 1 online resource (418 pages)
- Distribution:
- Ann Arbor : ProQuest Dissertations & Theses, 2022
- Contained In:
- Dissertations Abstracts International 84-01B.
- Place of Publication:
- [Philadelphia, Pennsylvania] : University of Pennsylvania, 2022.
- Language Note:
- English
- Summary:
- This thesis considers asymptotic behaviors of high-dimensional disordered systems, including Ising model and mean-field spin glass models. We first study the decay rate of correlations in the two-dimensional random field Ising model (RFIM). Second, we study the limit free energy of disordered systems. For RFIM, we are interested in the two-dimensional case where the external field is of i.i.d centered Gaussian variables. We show that under nonnegative temperature, the effect of boundary conditions on the magnetization in a finite box decays exponentially in the side length of the box. On the side of mean-field models, we use the Hamilton-Jacobi equation (HJE) approach, initiated by Jean-Christophe Mourrat, to characterize limiting free energy in many models from statistical inference problems and mean-field spin glass models. We now investigate infinite-dimensional models including many spin glass models and inference problems where the rank of the signal matrix increases as $n$ is sent to infinity. We give an intrinsic meaning to the Hamilton--Jacobi equation arising from mean-field spin glass models in the viscosity sense, and establish the corresponding well-posedness.This will shed more light on the mysterious Parisi formula as the limit of free energy in the Sherrington--Kirkpatrick model.
- Notes:
- Source: Dissertations Abstracts International, Volume: 84-01, Section: B.
- Advisors: Ding, Jian; Pemantle, Robin; Committee members: Sun, Xin.
- Department: Mathematics.
- Ph.D. University of Pennsylvania 2022.
- Local Notes:
- School code: 0175
- ISBN:
- 9798834092124
- Access Restriction:
- Restricted for use by site license.
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