## Doctoral Theses

### Scaling Limits of Some Random Interface Models.

9-22-2020

9-22-2021

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

#### Supervisor

Hazra, Rajat Subhra (TSMU-Kolkata; ISI)

#### Abstract (Summary of the Work)

In this thesis, we study some probabilistic models of random interfaces. Interfaces between different phases have been topic of considerable interest in statistical physics. These interfaces are described by a family of random variables, indexed by the ddimensional integer lattice, which are considered as a height configuration, namely they indicate the height of the interface above a reference hyperplane. The models are defined in terms of an energy function (Hamiltonian), which defines a Gibbs measure on the set of height configurations. More formally, letÏ• = {Ï•x}xâˆˆZ dbe a collection of real numbers indexed by the d-dimensional integer lattice Z d. Such a collection can be interpreted as a d-dimensional interface in d+ 1-dimensional Euclidean space R d+1 in the following manner: we think of Ï•x as height variable, indicating the height of the interface above the point x in the d-dimensional reference hyperplane. We obtain a d-dimensional surface in R d+1 by interpolating the heights linearly between the integer points. We will in general forget about the interpolation, and call any configuration {Ï•x}xâˆˆZ d an interface. We identify the family {Ï•x}xâˆˆZ d âˆˆ R Z d with the (graph of the) mappingÏ• : Zd â†’ Rsuch that Ï•(x) = Ï•x. We now introduce a probability measure on the set of interface configurations. Let â„¦ = R Z d be endowed with the product topology. We consider the product Ïƒ-field on â„¦. Let Î› be a finite subset of Z d . We fix a configuration {Ïˆx}xâˆˆZ d \\Î› which plays the role of a boundary condition. The probability of a configuration Ï• depends on its energy which is given by a Hamiltonian H Ïˆ Î› (Ï•). The probability measure on â„¦ is given (formally) byP Ïˆ,Î² Î› (dÏ•) := 1/Z Ïˆ,Î²Î› exp (âˆ’Î²HÏˆÎ› (Ï•) Ï€ xâˆˆÎ› dÏ•x Ï€ x /âˆˆÎ› Î´Ïˆx (dÏ•x)Here, Î² â‰¥ 0 is called the inverse temperature, dÏ•x is the one dimensional Lebesgue measure, Î´Ïˆx is the Dirac mass at Ïˆx and Z Ïˆ,Î²Î› is the constant which normalizes P Ïˆ,Î²Î› to a probability measure (if it is finite). In other words, if P Ïˆ,Î² Î› exists, it is the probability measure on the set of configurations restricted to be equal to Ïˆ outside Î› and has density (Z Ïˆ,Î² Î› ) âˆ’1 exp(âˆ’Î²HÏˆÎ› (Ï•)) with respect to the product Lebesgue measure on R Î›.Let us first see a concrete example of random interface models. The gradient model (or âˆ‡-model) is a random interface model, where the Hamiltonian is given byH Ïˆ Î› (Ï•) = 1/2 Î£x,yâˆˆÎ› px,yV (Ï•x âˆ’ Ï•y) + Î£xâˆˆÎ›,yâˆ‰Î› px,yV (Ï•x âˆ’ Ï•y).Here V : R â†’ R is an even convex function with V (0) = 0 and px,y is the transition matrix of a random walk on the lattice Z d . If we assume that the random walk has finite range, that is, the step distributions have finite support (there are more general conditions under which the measure is well defined), then (1.1.1) defines a probability measure on R Î›.

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28843866

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