Calculations are presented for a series of interrelated problems in the theory of disordered solids. The simple mean field theory of tricriticality in the layered Ising metamagnet is modified by inclusion of the Bethe-Peierls equation of state for the planar interactions. The approximation allows for a model of dilution with a finite percolation concentration for the layers. The calculated behaviour as a function of dilution for anisotropic coupling strengths allows comparison with experimental results on dilute ferrous chloride. Estimation is made of the effects of treating fluctuations by theories with mean field singularities. A discussion is given of first order phase transitions in disordered systems. A mean field theory of the implications of introducing quenched disorder to a system undergoing a transition of first order is effected by reformulating the problem in terms of a translationally invariant one via the replica method. The conclusions are examined in terms of a simple scaling theory and criteria derived for smearing of the singularities. Dynamic and thermodynamic properties of diluted magnetic insulators near the percolation concentration are considered in terms of simple geometric models of the percolating cluster as introduced by de Gennes. New scaling relations for the spin wave stiffness and conductivity exponents are derived and differences from previous relations interpreted geometrically. The scaling models of the percolating cluster so found, in which correlations propagate locally via effectively one-dimensional paths, are applied to the determination of the mobility edge for spin wave excitations in a dilute Heisenberg magnet near the percolation threshold. A prediction for the functional form of the mobility edge is made by means of results known for the Anderson model and a localisation length derived from an exact solution of a random problem by Dyson.

Thesis (M.S.)--University of Missouri--Rolla, 2007. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed March 27, 2008) Includes bibliographical references (p. 86-89).

Thesis (Ph.D.)--McMaster University (Canada), 1996. / Source: Dissertation Abstracts International, Volume: 57-10, Section: B, page: 6347. Adviser: N. Balakrishnan.

Thesis (Ph.D.)--Bowling Green State University, 2008. / Document formatted into pages; contains xii, 166 p. Includes bibliographical references.

Thesis (M.Sc.) - Carleton University, 2005. / Includes bibliographical references (p. 77-80). Also available in electronic format on the Internet.

Thesis (M.S.)--Ohio University, November, 1995. / Title from PDF t.p.

Thesis (Ph. D.)--Missouri University of Science and Technology, 2008. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed May 28, 2009) Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

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].

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