Studies in statistical mechanics and random geometry Zijie Zhuang

Format:

Book

Thesis/Dissertation

Author/Creator:

Zhuang, Zijie, author.

Contributor:

University of Pennsylvania. Statistics and Data Science., degree granting institution.

Language:

English

Subjects (All):

Mathematics.

Theoretical mathematics.

Statistics.

0405.

0642.

0463.

Local Subjects:

Mathematics.

Theoretical mathematics.

Statistics.

0405.

0642.

0463.

Genre:

Academic theses

Physical Description:

1 online resource (449 pages)

Contained In:

Dissertations Abstracts International 87-12B

Place of Publication:

Ann Arbor : ProQuest Dissertations and Theses, 2026

Language Note:

English

Summary:

This dissertation studies several problems at the interface of statistical mechanics and random geometry, with particular emphasis on critical planar percolation, Schramm-Loewner evolution (SLE), conformal loop ensembles (CLE), Liouville quantum gravity (LQG), log-correlated Gaussian fields, and disordered spin systems.The first part of the dissertation concerns critical planar Bernoulli percolation. It derives an exact expression for the backbone exponent, or monochromatic two-arm exponent, which governs the probability of two disjoint open crossings. In contrast to the previously known arm exponents, which are rational, this exponent is shown to be transcendental. The dissertation also establishes explicit annulus crossing formulae for three basic events in the continuum limit: the one-arm event, the polychromatic two-arm event, and the monochromatic two-arm event. The first two confirm predictions of Cardy, while the third yields a full expansion whose leading term is the backbone exponent and whose subleading terms reveal additional algebraic structure. These results rely on the relationship between percolation, CLE6, SLE bubble measures, conformal welding, and the exact solvability of Liouville conformal field theory. The second part studies exponential metrics associated with log-correlated Gaussian fields in arbitrary dimension. After suitable deterministic renormalization, tightness is proved in the subcritical regime, and every subsequential limit is shown to be a genuine metric inducing the Euclidean topology. This extends a fundamental step in the construction of LQG-type metrics beyond the planar setting, where conformal invariance and Russo-Seymour-Welsh methods are unavailable, and instead requires multiscale geometric and percolative arguments.The third part addresses the conformal removability of non-simple SLE. It is shown that, for κ sufficiently close to 8, the trace of SLEκ contains a topological Sierpiński carpet. Combined with known results on such carpets, this implies conformal non-removability of SLEκ in this regime. The argument is based on a comparison with fractal percolation and on the analysis of the adjacency graph of complementary connected components, revealing a phase transition in the geometry of SLE traces within the interval (4, 8). The final part concerns disordered spin systems. The dissertation gives a new proof of long-range order for the random field Ising model in dimension d ≥ 3 at low temperature and weak disorder, and extends the method to random field Potts models. The proof replaces earlier renormalization group arguments with a robust Peierls-type strategy that simultaneously flips spins and disorder in a region and controls the corresponding change in the partition function. This framework also yields further consequences, including lower bounds on the correlation length in two dimensions.Together, these results develop robust new tools and resolve open problems in both fields, yielding new insights into statistical mechanics and random geometry.

Notes:

Source: Dissertations Abstracts International, Volume: 87-12, Section: B.

Advisors: Bhattacharya, Bhaswar B. Committee members: Ding, Jian; Huang, Jiaoyang; Sun, Xin

Ph.D. University of Pennsylvania 2026

Vendor supplied data

Local Notes:

School code: 0175

ISBN:

9798247973348

Access Restriction:

Restricted for use by site license