Slow Mixing of Glauber Dynamics for the Six-Vertex Model in the Ordered Phases

The six-vertex model in statistical physics is a weighted generalization of the ice model on Z^2 (i.e., Eulerian orientations) and the zero-temperature three-state Potts model (i.e., proper three-colorings). The phase diagram of the model represents its physical properties and suggests where local Markov chains will be efficient. In this paper, we analyze the mixing time of Glauber dynamics for the six-vertex model in the ordered phases. Specifically, we show that for all Boltzmann weights in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the first rigorous result bounding the mixing time of Glauber dynamics in the ferroelectric phase. Our analysis demonstrates a fundamental connection between correlated random walks and the dynamics of intersecting lattice path models (or routings). We analyze the Glauber dynamics for the six-vertex model with free boundary conditions in the antiferroelectric phase and significantly extend the region for which local Markov chains are known to be slow mixing. This result relies on a Peierls argument and novel properties of weighted non-backtracking walks

Torpid Mixing of Markov Chains for the Six-vertex Model on Z^2

In this paper, we study the mixing time of two widely used Markov chain algorithms for the six-vertex model, Glauber dynamics and the directed-loop algorithm, on the square lattice Z^2. We prove, for the first time that, on finite regions of the square lattice these Markov chains are torpidly mixing under parameter settings in the ferroelectric phase and the anti-ferroelectric phase

Probabilistic techniques for analyzing sampling algorithms and the dynamics of lattice models

Statistical mechanics bridges the fields of physics and probability theory, providing critical insights into both disciplines. Statistical physics models capture key features of macroscopic phenomena and consist of a set of configurations satisfying various constraints. Markov chain Monte Carlo algorithms are often used to sample from distributions over the exponentially large state space of these models to gain insight about the system and estimate its thermodynamic properties. Similar problems arise throughout machine learning, optimization, and counting complexity. In this dissertation, we present several new techniques based on random walks for analyzing sampling algorithms and the dynamics of various lattice models from statistical physics. We start by investigating the mixing time of Glauber dynamics for the six-vertex model in its ordered phases. We show that for every Boltzmann weight in the ferroelectric phase, there exist boundary conditions such that local Markov chains require exponential time to converge to equilibrium. This is the first rigorous result about the mixing time of Glauber dynamics for the six-vertex model in the ferroelectric phase. We also analyze the Glauber dynamics with free boundary conditions in the antiferroelectric phase and significantly extend the region for which local Markov chains are known to be slow mixing. In separate lines of work, we use techniques from the theory of random walks and electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model, closing an open problem of Babai and Gorodezky. The Abelian sandpile model is the canonical dynamical system used to study the phenomenon of self-organized criticality, and the transience class measures the time needed for the process to reach steady-state behavior. We also explore a new approach for approximately sampling elements with fixed rank from graded posets that relies solely on the mixing time of biased Markov chains. This allows us to bypass the usual obstacle of log-concavity. Last, we take a foray into analytic combinatorics and use the singularity analysis of Dirichlet generating functions to design sampling algorithms for Bose–Einstein condensates.Ph.D

The communicative theory of Terminology (CTT) applied to the development of a corpus-based specialised dictionary of the ceramics industry

Esta tesis es el resultado de un proyecto destinado a la creación de un diccionario activo, bilingüe (español-inglés; inglés-español) y especializado de la industria cerámica y azulejera con la Teoría Comunicativa de la Terminología como su pilar teórico principal. Debido al posicionamiento teórico adoptado, la investigación aquí presentada ha partido de un estudio de corpus (compilado ad hoc) en el que los términos han sido analizados in vivo y caracterizados de acuerdo al ¿habitat¿ en el que se hallan en el texto especializado. Así pues, la aproximación hecha al estudio de la terminología industrial cerámica hace pertinente el uso de la etiqueta ¿lexicografía especializada¿ a la hora de referirnos a un trabajo como éste en el que se ha tratado de ir más allá de la práctica terminográfica para dar lugar a un estudio en el que se prima el contexto, las asociaciones naturales de los términos (colocaciones) y la naturaleza comunicativa de la terminología. De este modo, en esta tesis se ha presentado de manera progresiva, además de un marco teórico detallado y coherente con el fin último de la investigación, la metodología utilizada para la elaboración del diccionario en curso, ampliamente basada en el uso de programas informáticos tanto para la explotación del corpus (WordSmith Tools 4.0), como para la creación de la base de datos terminológica (TermStar XV) y la generación de entradas finales (GENDIC).Así pues, esta tesis presenta de manera progresiva los resultados obtenidos en cada etapa del método de trabajo y 4,000 entradas finales (en este caso del inglés al español) correspondientes a las letras A, B, N, O, U y V del diccionario.This PhD dissertation is the result of an ongoing process aimed at the creation of a bilingual corpus-based specialised active dictionary of the ceramic industry, with the Communicative Theory of Terminology (CTT) as its mainstay. According to the grounding principles of the CTT, this research has departed form a corpus-based approach in which terms have been analysed in vivo and characterised from the natural habitat in which they are given in specialised communication/discourse. In this light, it has been put forward how the study of terms – made possible thanks to the activity of compiling and describing them, called terminography – may be complemented by the wider projection of specialised lexicography for the compilation and elaboration of LSP, user-oriented and user-friendly quality products in the form of dictionaries. This specialised lexicographical dimension of the work has necessarily implied the need to renew the concept of speciality language dictionaries applied to the ceramic industry and has given way to the creation of a (prospective) active dictionary in this field with a marked emphasis on context. Accordingly, the importance of pragmatic aspects in a work of this sort, has made it necessary to undertake an in-depth revision and analysis of the socio-economic context for the research in order be able to establish and solve the specific terminological needs that the ceramic industrial discourse community may find. On the basis of this theoretical framework, the method of study followed for the development of the prospective dictionary has comprised 8 broad stages: the stage of work preparation and corpus compilation, the elaboration of the field diagram, the stage of documentary corpus management, term extraction, data processing, revision and normalisation and finally, the edition stage. Two main types of results have been presented: those obtained through work in progress in the different stages of the method and final ones strictly speaking, that is, 4,000 English-Spanish entries in their final format (as they will appear in the prospective dictionary) belonging to the letters A, B, N, O, U and V of a complete dictionary which will include a total of 26,000 entries

The communicative theory of Terminology (CTT) applied to the development of a corpus-based specialised dictionary of the ceramics industry

Esta tesis es el resultado de un proyecto destinado a la creación de un diccionario activo, bilingüe (español-inglés; inglés-español) y especializado de la industria cerámica y azulejera con la Teoría Comunicativa de la Terminología como su pilar teórico principal. Debido al posicionamiento teórico adoptado, la investigación aquí presentada ha partido de un estudio de corpus (compilado ad hoc) en el que los términos han sido analizados in vivo y caracterizados de acuerdo al ¿habitat¿ en el que se hallan en el texto especializado. Así pues, la aproximación hecha al estudio de la terminología industrial cerámica hace pertinente el uso de la etiqueta ¿lexicografía especializada¿ a la hora de referirnos a un trabajo como éste en el que se ha tratado de ir más allá de la práctica terminográfica para dar lugar a un estudio en el que se prima el contexto, las asociaciones naturales de los términos (colocaciones) y la naturaleza comunicativa de la terminología. De este modo, en esta tesis se ha presentado de manera progresiva, además de un marco teórico detallado y coherente con el fin último de la investigación, la metodología utilizada para la elaboración del diccionario en curso, ampliamente basada en el uso de programas informáticos tanto para la explotación del corpus (WordSmith Tools 4.0), como para la creación de la base de datos terminológica (TermStar XV) y la generación de entradas finales (GENDIC).Así pues, esta tesis presenta de manera progresiva los resultados obtenidos en cada etapa del método de trabajo y 4,000 entradas finales (en este caso del inglés al español) correspondientes a las letras A, B, N, O, U y V del diccionario.This PhD dissertation is the result of an ongoing process aimed at the creation of a bilingual corpus-based specialised active dictionary of the ceramic industry, with the Communicative Theory of Terminology (CTT) as its mainstay. According to the grounding principles of the CTT, this research has departed form a corpus-based approach in which terms have been analysed in vivo and characterised from the natural habitat in which they are given in specialised communication/discourse. In this light, it has been put forward how the study of terms – made possible thanks to the activity of compiling and describing them, called terminography – may be complemented by the wider projection of specialised lexicography for the compilation and elaboration of LSP, user-oriented and user-friendly quality products in the form of dictionaries. This specialised lexicographical dimension of the work has necessarily implied the need to renew the concept of speciality language dictionaries applied to the ceramic industry and has given way to the creation of a (prospective) active dictionary in this field with a marked emphasis on context. Accordingly, the importance of pragmatic aspects in a work of this sort, has made it necessary to undertake an in-depth revision and analysis of the socio-economic context for the research in order be able to establish and solve the specific terminological needs that the ceramic industrial discourse community may find. On the basis of this theoretical framework, the method of study followed for the development of the prospective dictionary has comprised 8 broad stages: the stage of work preparation and corpus compilation, the elaboration of the field diagram, the stage of documentary corpus management, term extraction, data processing, revision and normalisation and finally, the edition stage. Two main types of results have been presented: those obtained through work in progress in the different stages of the method and final ones strictly speaking, that is, 4,000 English-Spanish entries in their final format (as they will appear in the prospective dictionary) belonging to the letters A, B, N, O, U and V of a complete dictionary which will include a total of 26,000 entries