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1.
Li, Zhongyang.
Vertex Models, Ising Models and Fisher Graphs.
Degree: PhD, Mathematics, 2011, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11314/ 
► In Chapter 1, we study planar ``vertex'' models, which are probability measures onedge subsets of a planar graph, satisfying certain constraints ateach vertex, examples including…
(more)
▼ In Chapter 1, we study planar ``vertex'' models, which
are probability measures onedge subsets of a planar graph,
satisfying certain constraints ateach vertex, examples including
dimer model, and 1-2 model, which wewill define. We express the
local statistics of a large class ofvertex models on a finite
hexagonal lattice as a linear combinationof the local statistics of
dimers on the corresponding Fisher graph,with the help of a
generalized holographic algorithm. Using ann × n torus to
approximate the periodic infinite graph, wegive an explicit
integral formula for the free energy and localstatistics for
configurations of the vertex model on an infinitebi-periodic graph.
As an example, we simulate the 1-2 model by thetechnique of Glauber
dynamics. In Chapter 2, we study the spectral curves of dimer
models on periodic Fishergraphs, defined by the zero locus of the
determinant of a modifiedweighted adjacency matrix. We prove that
either they are disjointfrom the unit torus
(\mathbb{T}
2={(z,w):|z|=1,|w|=1}) or theyintersect
\mathbb{T}
2 at a single real point. As an application, we prove
that the single edge probability of dimer models on periodic Fisher
graphs is unique under any translation invariant Gibbs measure. A
periodic Ising model is one endowed with interactions that
areinvariant under translations of members of a full-rank
sublattice\mathfrak{L} of ℤ
2. In Chapter 3, we give
an exact, quantitativedescription of the critical temperature,
defined by the supreme ofthe temperatures at which the spontaneous
magnetization of aperiodic, Ising ferromagnets is nonzero, as the
solution of acertain algebraic equation, namely, the condition that
the spectralcurve of the corresponding dimer model on the Fisher
graph has areal node on the unit torus. A simple proof for the
exponentialdecay of spin-spin correlations above the critical
temperature forthe symmetric, periodic Ising ferromagnet, as well
as theexponential decay of the edge-edge correlations for all
non-criticaledge weights of the dimer model on periodic Fisher
graphs, isobtained by our technique.
Advisors/Committee Members: Kenyon, Richard (Director), Kenyon, Richard (Reader), Ramanan, Kavita (Reader), Wilson, David (Reader).
Subjects/Keywords: Local Statistics
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APA (6th Edition):
Li, Z. (2011). Vertex Models, Ising Models and Fisher Graphs. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11314/
Chicago Manual of Style (16th Edition):
Li, Zhongyang. “Vertex Models, Ising Models and Fisher Graphs.” 2011. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:11314/.
MLA Handbook (7th Edition):
Li, Zhongyang. “Vertex Models, Ising Models and Fisher Graphs.” 2011. Web. 16 Jan 2021.
Vancouver:
Li Z. Vertex Models, Ising Models and Fisher Graphs. [Internet] [Doctoral dissertation]. Brown University; 2011. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:11314/.
Council of Science Editors:
Li Z. Vertex Models, Ising Models and Fisher Graphs. [Doctoral Dissertation]. Brown University; 2011. Available from: https://repository.library.brown.edu/studio/item/bdr:11314/
2.
Le, Quang Nhat.
A family of projectively natural polygon iterations.
Degree: Department of Mathematics, 2017, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:733400/
► Polygon iterations, which can be thought of as discrete dynamical systems on the space of polygons, provide an abundance of interesting discrete dynamical systems in…
(more)
▼ Polygon iterations, which can be thought of as
discrete dynamical systems on the space of polygons, provide an
abundance of interesting discrete dynamical systems in geometry,
especially in Euclidean and affine geometries. Recently, the
advance of computers has allowed the study of polygon iterations in
projective geometry, which was previously limited by the high
computational complexity of the associated rational maps, to take
off. Notable examples are the pentagram map and the projective
midpoint map, both first studied by
Richard Schwartz as potential
analogues of the classical midpoint map. In this thesis, we will
investigate a one-parameter family of projectively natural polygon
iterations that includes both the pentagram map and the projective
midpoint map. They can be regarded as autonomous discrete dynamical
systems on the non-compact space of polygons, modulo projective
transformations. Except for two parameters, corresponding to the
pentagram map and its inverse, these polygon iterations are
observed to possess a single globally attracting fixed point, which
allows us to define their Julia sets. Coincidentally, when
observing the varying Julia sets, we discovered that this family
contains two projective analogues of Varignon's theorem for
quadrilaterals.
Advisors/Committee Members: Schwartz, RIchard (Advisor), Kenyon, RIchard (Reader), Robins, Sinai (Reader).
Subjects/Keywords: Dynamics
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APA (6th Edition):
Le, Q. N. (2017). A family of projectively natural polygon iterations. (Thesis). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:733400/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Le, Quang Nhat. “A family of projectively natural polygon iterations.” 2017. Thesis, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:733400/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Le, Quang Nhat. “A family of projectively natural polygon iterations.” 2017. Web. 16 Jan 2021.
Vancouver:
Le QN. A family of projectively natural polygon iterations. [Internet] [Thesis]. Brown University; 2017. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:733400/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Le QN. A family of projectively natural polygon iterations. [Thesis]. Brown University; 2017. Available from: https://repository.library.brown.edu/studio/item/bdr:733400/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
3.
Gokturk, Ali.
Comparison of Teichmüller geodesics and Weil-Petersson
geodesics.
Degree: PhD, Mathematics, 2011, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11309/
► Let S be a surface with genus g and n punctures and let x(S)=3g+n denote the complexity of the surface S. We prove that in…
(more)
▼ Let S be a surface with genus g and n punctures and
let x(S)=3g+n denote the complexity of the surface S. We prove that
in the Teichmüller space T(S) endowed with the Weil-Petersson
metric, every Teichmüller geodesics segment fellow-travels the
Weil-Petersson geodesic segment with the same pair of end points if
and only if x(S)=5.
Advisors/Committee Members: Brock, Jeffrey (Director), Schwartz, Richard (Reader), Kenyon, Richard (Reader).
Subjects/Keywords: Teichm�metric
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APA (6th Edition):
Gokturk, A. (2011). Comparison of Teichmüller geodesics and Weil-Petersson
geodesics. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11309/
Chicago Manual of Style (16th Edition):
Gokturk, Ali. “Comparison of Teichmüller geodesics and Weil-Petersson
geodesics.” 2011. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:11309/.
MLA Handbook (7th Edition):
Gokturk, Ali. “Comparison of Teichmüller geodesics and Weil-Petersson
geodesics.” 2011. Web. 16 Jan 2021.
Vancouver:
Gokturk A. Comparison of Teichmüller geodesics and Weil-Petersson
geodesics. [Internet] [Doctoral dissertation]. Brown University; 2011. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:11309/.
Council of Science Editors:
Gokturk A. Comparison of Teichmüller geodesics and Weil-Petersson
geodesics. [Doctoral Dissertation]. Brown University; 2011. Available from: https://repository.library.brown.edu/studio/item/bdr:11309/
4.
Healey, Vivian Olsiewski.
The Loewner Equation with Branching and the Continuum Random
Tree.
Degree: Department of Mathematics, 2017, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:733356/
► The present work brings together the fields of random maps and Loewner evolution by constructing explicit embeddings of critical Galton-Watson trees in the upper half-plane…
(more)
▼ The present work brings together the fields of random
maps and Loewner evolution by constructing explicit embeddings of
critical Galton-Watson trees in the upper half-plane via the
Loewner equation and considering the scaling limit of the
associated time-dependent random driving measures as the finite
trees converge to the continuum random tree. Chapter 2 addresses
the (deterministic) conformal mapping problem of incorporating
branching into the Loewner equation. We identify sufficient
conditions on the driving measure for the Loewner equation to
generate a union of two simple curves that meet at a fixed
nontrivial angle on the real line, which is the fundamental step in
generating graph embeddings of trees. Chapter 3 identifies a
specific repulsive force (the deterministic part of Dyson Brownian
motion) that, when used to describe the evolution of a random
discrete measure whose atoms represent the particles of a
Galton-Watson branching process, satisfies the conditions for tree
embedding given in Chapter 2. Chapter 4 investigates the scaling
limit of these time-dependent driving measures through the lens of
superprocesses. In the setting when the critical Galton-Watson
trees are conditioned to converge to the continuum random tree, the
sequence of measure-valued processes is shown to be tight. In order
to identify the limit, the question of convergence of the sequence
of measures is reframed as a question concerning the associated
sequence of Stieltjes transforms. For each measure-valued process
in the sequence, the flow of the associated Stieltjes transform is
shown to satisfy a particular SPDE that is related to the complex
Burgers equation. Finally, in the unconditioned case, the density ρ
of the limiting superprocess is conjectured to satisfy the equation
∂tρ + ∂x (ρ · Hρ) = σ√ρ · W , where H is the Hilbert transform, W
is space-time white noise, and σ is a positive
constant.
Advisors/Committee Members: Menon, Govind (Advisor), Kenyon, Richard (Reader), Rohde, Steffen (Reader).
Subjects/Keywords: Geometric function theory
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APA (6th Edition):
Healey, V. O. (2017). The Loewner Equation with Branching and the Continuum Random
Tree. (Thesis). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:733356/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Healey, Vivian Olsiewski. “The Loewner Equation with Branching and the Continuum Random
Tree.” 2017. Thesis, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:733356/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Healey, Vivian Olsiewski. “The Loewner Equation with Branching and the Continuum Random
Tree.” 2017. Web. 16 Jan 2021.
Vancouver:
Healey VO. The Loewner Equation with Branching and the Continuum Random
Tree. [Internet] [Thesis]. Brown University; 2017. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:733356/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Healey VO. The Loewner Equation with Branching and the Continuum Random
Tree. [Thesis]. Brown University; 2017. Available from: https://repository.library.brown.edu/studio/item/bdr:733356/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
5.
Yi, Ren.
Domain Exchange Transformations and Their
Renormalizations.
Degree: Department of Mathematics, 2018, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:792737/
► Domain exchange transformations (DETs) are dynamical systems of piecewise translations defined on piecewise smooth Jordan domains. DETs are two-dimensional generalizations of interval exchange transformations which…
(more)
▼ Domain exchange transformations (DETs) are dynamical
systems of piecewise translations defined on piecewise smooth
Jordan domains. DETs are two-dimensional generalizations of
interval exchange transformations which have been
fruitfully-studied for more than 40 years. For DETs,
renormalizations are the key to describe complete dynamical
behaviors. We say that a DET is renormalizable if there is a
subdomain to which the first return map is conjugate to the DET
itself by an affine map. In this thesis, we study DETs arising from
two general constructions and their renormalizations. Polygon
exchange transformations (PETs) are DETs defined on polygonal
domains with partitions into subpolygons. One general method of
constructing PETs based on multigraphs was introduced by R.
Schwartz in 2013. The resulting dynamical systems are called the
multigraph PETs. In the first part of this thesis, we describe a
1-parameter family of multigraph PETs called the triple lattice
PETs. We show that there exists a renormalization scheme of the
triple lattice PETs in the parameter space (0, 1). By the
renormalization scheme, we are able to analyze the limit set Λ when
the parameter is the golden ratio. We show that Λ is the nested
intersection of a countable sequence of finite unions of isosceles
trapezoids and its Hausdorff dimension is 1.83157… which is less
than 2. In the second part of this thesis, we explain how to use
cut-and-and project set to construct minimal DETs. Specializing to
the case when the domain is a unit square and the cut-and-project
set is associated to a Galois lattice, we construct an infinite
family of RETs (DETs on the unit square partitioned into
rectilinear polygons). Each RET in the family is called a PV RET
since every map is associated to a PV number. We prove that all PV
RETs in the family are renormalizable. Moreover, we discover that
certain PV RETs can be composed to create a new RET with the same
combinatorics as any PV RET. The composition is called an
admissible RET. We find a family of admissible RETs, called
multistage RETs, which are renormalizable. This is a joint work
with Ian Alevy and
Richard Kenyon.
Advisors/Committee Members: Schwartz, Richard (Advisor), Kenyon, Richard (Reader), Brock, Jeffrey (Reader).
Subjects/Keywords: dynamical systems
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APA (6th Edition):
Yi, R. (2018). Domain Exchange Transformations and Their
Renormalizations. (Thesis). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:792737/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Yi, Ren. “Domain Exchange Transformations and Their
Renormalizations.” 2018. Thesis, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:792737/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Yi, Ren. “Domain Exchange Transformations and Their
Renormalizations.” 2018. Web. 16 Jan 2021.
Vancouver:
Yi R. Domain Exchange Transformations and Their
Renormalizations. [Internet] [Thesis]. Brown University; 2018. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:792737/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Yi R. Domain Exchange Transformations and Their
Renormalizations. [Thesis]. Brown University; 2018. Available from: https://repository.library.brown.edu/studio/item/bdr:792737/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
6.
Davis, Diana.
Cutting Sequences on Translation Surfaces.
Degree: PhD, Mathematics, 2013, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:320626/
► We analyze the cutting sequences associated to geodesic flow on a large class of translation surfaces, including Bouw-Möller surfaces and certain rectilinear surfaces. We give…
(more)
▼ We analyze the cutting sequences associated to
geodesic flow on a large class of translation surfaces, including
Bouw-Möller surfaces and certain rectilinear surfaces. We give a
combinatorial rule that relates a cutting sequence corresponding to
a given trajectory, to the cutting sequence corresponding to the
image of that trajectory under the parabolic element of the Veech
group. This extends previous work for regular polygon surfaces to a
larger class of translation surfaces. We find that the
combinatorial rule is the same as for regular polygon surfaces in
about half of the cases, and different in the other
half.
Advisors/Committee Members: Schwartz, Richard (Director), Brock, Jeffrey (Reader), Kenyon, Richard (Reader).
Subjects/Keywords: translation surfaces
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APA (6th Edition):
Davis, D. (2013). Cutting Sequences on Translation Surfaces. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:320626/
Chicago Manual of Style (16th Edition):
Davis, Diana. “Cutting Sequences on Translation Surfaces.” 2013. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:320626/.
MLA Handbook (7th Edition):
Davis, Diana. “Cutting Sequences on Translation Surfaces.” 2013. Web. 16 Jan 2021.
Vancouver:
Davis D. Cutting Sequences on Translation Surfaces. [Internet] [Doctoral dissertation]. Brown University; 2013. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:320626/.
Council of Science Editors:
Davis D. Cutting Sequences on Translation Surfaces. [Doctoral Dissertation]. Brown University; 2013. Available from: https://repository.library.brown.edu/studio/item/bdr:320626/
7.
Newkirk, Edward S.
Billiards with Bombs.
Degree: PhD, Mathematics, 2016, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:674241/
► In this article, we define a variant of billiards in which the ball bounces around a square grid erasing walls as it goes. We prove…
(more)
▼ In this article, we define a variant of billiards in
which the ball bounces around a square grid erasing walls as it
goes. We prove that there exist periodic tunnels with arbitrarily
large period from any possible starting point, that there exist
nonperiodic tunnels from any possible starting point, and that
there are versions of the problem for which the same starting point
and initial direction result in periodic tunnels of arbitrarily
large period. We conjecture that there exist starting conditions
which do not lead to tunnels, justify the conjecture with
simulation evidence, and discuss the difficulty of proving
it.
Advisors/Committee Members: Schwartz, Richard (Director), Kenyon, Richard (Reader), Brock, Jeffrey (Reader).
Subjects/Keywords: cutting sequence
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APA (6th Edition):
Newkirk, E. S. (2016). Billiards with Bombs. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:674241/
Chicago Manual of Style (16th Edition):
Newkirk, Edward S. “Billiards with Bombs.” 2016. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:674241/.
MLA Handbook (7th Edition):
Newkirk, Edward S. “Billiards with Bombs.” 2016. Web. 16 Jan 2021.
Vancouver:
Newkirk ES. Billiards with Bombs. [Internet] [Doctoral dissertation]. Brown University; 2016. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:674241/.
Council of Science Editors:
Newkirk ES. Billiards with Bombs. [Doctoral Dissertation]. Brown University; 2016. Available from: https://repository.library.brown.edu/studio/item/bdr:674241/
8.
Alevy, Ian.
Regular Polygon Surfaces and Renormalizable Rectangle
Exchange Maps.
Degree: Department of Applied Mathematics, 2018, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:792695/
► We investigate two different topics in discrete mathematics: the geometry of piecewise-linear surfaces and the long-term behavior of discrete dynamical systems. A regular polygon surface…
(more)
▼ We investigate two different topics in discrete
mathematics: the geometry of piecewise-linear surfaces and the
long-term behavior of discrete dynamical systems. A regular polygon
surface M is a surface graph (Sigma, Gamma) together with a
continuous map psi from Sigma into Euclidean 3-space which maps
faces to regular Euclidean polygons. When Sigma is homeomorphic to
the sphere, and the degree of every face of Gamma is five, we prove
that M can be realized as the boundary of a union of dodecahedra
glued together along common facets. Under the same assumptions but
when the faces of Gamma have degree four or eight, we prove that M
can be realized as the boundary of a union of cubes and octagonal
prisms glued together along common facets. We exhibit
counterexamples showing the failure of both theorems for higher
genus surfaces. In joint work with
Richard Kenyon and Ren Yi we
study domain exchange maps (DEMs). A DEM is a dynamical system
defined on a smooth Jordan domain which is a piecewise translation.
We explain how to use cut-and-project sets to construct minimal
DEMs. Specializing to the case in which the domain is a square and
the cut-and-project set is associated to a Galois lattice, we
construct an infinite family of DEMs in which each map is
associated to a PV number. We develop a renormalization scheme for
these DEMs. Certain DEMs in the family can be composed to create
multistage, renormalizable DEMs.
Advisors/Committee Members: Kenyon, Richard (Advisor), Menon, Govind (Reader), Schwartz, Richard (Reader).
Subjects/Keywords: Dynamics
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APA ·
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to Zotero / EndNote / Reference
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APA (6th Edition):
Alevy, I. (2018). Regular Polygon Surfaces and Renormalizable Rectangle
Exchange Maps. (Thesis). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:792695/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Chicago Manual of Style (16th Edition):
Alevy, Ian. “Regular Polygon Surfaces and Renormalizable Rectangle
Exchange Maps.” 2018. Thesis, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:792695/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
MLA Handbook (7th Edition):
Alevy, Ian. “Regular Polygon Surfaces and Renormalizable Rectangle
Exchange Maps.” 2018. Web. 16 Jan 2021.
Vancouver:
Alevy I. Regular Polygon Surfaces and Renormalizable Rectangle
Exchange Maps. [Internet] [Thesis]. Brown University; 2018. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:792695/.
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
Council of Science Editors:
Alevy I. Regular Polygon Surfaces and Renormalizable Rectangle
Exchange Maps. [Thesis]. Brown University; 2018. Available from: https://repository.library.brown.edu/studio/item/bdr:792695/
Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation
9.
tassy, martin.
Tilings by bars.
Degree: PhD, Mathematics, 2014, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:386276/
► We study combinatorial and probabilistic properties of tilings of the plane by m × 1 horizontal rectangles and n × 1 vertical rectangles also called tilings by bars. In…
(more)
▼ We study combinatorial and probabilistic properties of
tilings of the plane by m × 1 horizontal rectangles and
n × 1 vertical rectangles also called tilings by bars. In
Chapter 2 We give a new criterion for the tilability of regions
based on the height function in the Conway group. As a consequence
we are able to prove that the tilability or regions with bounded
number of holes and such that the boundary of those holes describes
a trivial word in the Conway group is decidable in polynomial time.
In the second part of Chapter 2 we generalize results on local
moves connectivity known for simply connected regions by giving a
criterion to decide under which conditions tilings of a torus are
connected by local moves. In Chapter 3 we discuss the differences
between the space of ergodic Gibbs measures for tilings by dominos
and the space of ergodic Gibbs measures for tilings by longer bars.
Using notions from ergodic theory we prove that when a measure
μ on tiling of the plane is ergodic then the image of the
height function must μ -almost surely stay close to a single
geodesic. We also propose a characterization of those ergodic Gibbs
measures based on the generalization of the domino slope and on the
support of the height function.
Advisors/Committee Members: Kenyon, Richard (Director), Schwartz, Richard (Reader), Brock, Jeffrey (Reader).
Subjects/Keywords: Tilings
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APA (6th Edition):
tassy, m. (2014). Tilings by bars. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:386276/
Chicago Manual of Style (16th Edition):
tassy, martin. “Tilings by bars.” 2014. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:386276/.
MLA Handbook (7th Edition):
tassy, martin. “Tilings by bars.” 2014. Web. 16 Jan 2021.
Vancouver:
tassy m. Tilings by bars. [Internet] [Doctoral dissertation]. Brown University; 2014. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:386276/.
Council of Science Editors:
tassy m. Tilings by bars. [Doctoral Dissertation]. Brown University; 2014. Available from: https://repository.library.brown.edu/studio/item/bdr:386276/
10.
Ma, Ningning.
Tropicalization of the Dimer Model.
Degree: PhD, Mathematics, 2015, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:419470/
► In this paper, we study the the tropical (freezing) behavior of the dimer model and the cluster integrable system arising from it. The dimer model…
(more)
▼ In this paper, we study the the tropical (freezing)
behavior of the dimer model and the cluster integrable system
arising from it. The dimer model is an example of the two
dimensional statistical mechanics model. The partition function and
local statistics of the dimer model can be computed explicitly. And
the phase diagram of the Gibbs measure for dimers on a graph G is
represented by the amoeba of an associated plane algebraic curve,
the spectral curve. We are interested in the macroscopic behavior
of the system when the temperature T goes to zero. We show that a
frozen phase is given for almost all ergodic Gibbs measures, except
those on the so-called tropical amoeba. In Chapter 3, we study the
tropical amoeba arising from a given dimer model, and prove that
when the ergodic Gibbs measure is given by a point on the tropical
amoeba, then generically the system will have the similar behavior
as the uniform dimer model on a honeycomb graph with some specified
external magnetic field. The dimer model also has some
non-probabilistic applications. In particular, Goncharov and
Kenyon
show that the dimer model on a periodic, bipartite graph gives rise
to a so-called cluster integrable system. Applying the
tropicalization process, we obtain a tropical integrable system,
that is, a system given by piecewise-linear equations. The study of
this system might be a nice application of tropical geometry. In
Chapter 4, we present concrete computation in the cases of genus
one and two, provide the conjecture that the general isolevel set
is always isomorphic to the tropical Jacobian of the spectral
curve, and give a discussion of the time evolution of the
systems.
Advisors/Committee Members: Kenyon, Richard (Director), Abramovich, Dan (Reader), Klivans, Caroline (Reader).
Subjects/Keywords: dimer model
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APA ·
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MLA ·
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APA (6th Edition):
Ma, N. (2015). Tropicalization of the Dimer Model. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:419470/
Chicago Manual of Style (16th Edition):
Ma, Ningning. “Tropicalization of the Dimer Model.” 2015. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:419470/.
MLA Handbook (7th Edition):
Ma, Ningning. “Tropicalization of the Dimer Model.” 2015. Web. 16 Jan 2021.
Vancouver:
Ma N. Tropicalization of the Dimer Model. [Internet] [Doctoral dissertation]. Brown University; 2015. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:419470/.
Council of Science Editors:
Ma N. Tropicalization of the Dimer Model. [Doctoral Dissertation]. Brown University; 2015. Available from: https://repository.library.brown.edu/studio/item/bdr:419470/
11.
Chhita, Sunil.
Scaling Windows of Dimer Models.
Degree: PhD, Mathematics, 2011, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11292/
► A scaling window is a type of continuum measure for a grid based statistical mechanical model, obtained when the grid size tends to zero. Contrary…
(more)
▼ A scaling window is a type of continuum measure for a
grid based statistical mechanical model, obtained when the grid
size tends to zero. Contrary to scaling limits, scaling windows are
often non-trivial for off-critical statistical mechanical models.
This thesis focuses on the scaling windows of dimer models which
are simplified models for the absorption of diatomic molecules. We
study the dimer model on two particular graphs � the so-called
Fisher lattice and the square grid.A natural parameterization of
the dimer model on the Fisher lattice gives a one dimensional
particle system equipped with creations and annihilations. It turns
out that the model is in two-to-one correspondence with an
anti-ferromagnetic anisotropic Ising model. We can compute the
exact phase diagram and locate two values of interest � the
critical value and the independent value. At criticality, under a
certain choice of scaling window which happens to give dense sets
of creations and annihilations, we find that the stationary
distribution is a Pfaffian point process. We also find that the
distribution of creations giving macroscopic paths is a Poisson
point process. At the independent value, we find that the particle
system is in bijection with a noisy voter model whose scaling
window is already known and is given by the Continuum Noisy Voter
Model.The dimer model on a planar bipartite graph can be viewed as
a random surface measure. We study these fluctuations for dimer
models on the square grid equipped with two different classes of
weights, named flipped and drifted. In fact, the height
fluctuations for these two measures are equivalent. We show that
the height fluctuations of the dimer model with flipped weights are
non-Gaussian in the scaling window.
Advisors/Committee Members: Kenyon, Richard (Director), Menon, Govind (Reader), Boutillier, C�ic (Reader).
Subjects/Keywords: dimer model
Record Details
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Cite
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Record Details
Similar Records
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Chhita, S. (2011). Scaling Windows of Dimer Models. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11292/
Chicago Manual of Style (16th Edition):
Chhita, Sunil. “Scaling Windows of Dimer Models.” 2011. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:11292/.
MLA Handbook (7th Edition):
Chhita, Sunil. “Scaling Windows of Dimer Models.” 2011. Web. 16 Jan 2021.
Vancouver:
Chhita S. Scaling Windows of Dimer Models. [Internet] [Doctoral dissertation]. Brown University; 2011. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:11292/.
Council of Science Editors:
Chhita S. Scaling Windows of Dimer Models. [Doctoral Dissertation]. Brown University; 2011. Available from: https://repository.library.brown.edu/studio/item/bdr:11292/
. 
Open Access Theses and Dissertations