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 Events
Tyler Helmuth
Tue 25 Nov 2014, 4:00pm SPECIAL
One Time Event
Graduate Student Center, Room 203
Doctoral Exam
Graduate Student Center, Room 203
Tue 25 Nov 2014, 4:00pm-6:30pm

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Spin systems such as the Ising model are central topics in statistical mechanics and probability theory. In the late 1960s Symanzik made the important discovery that properties of spin systems could be expressed in terms of the behaviour of random walks. This thesis contributes to the understanding of these connections by developing and analyzing random walk representations of graphical models arising in statistical mechanics.
Concretely, the results of this thesis can be divided into two parts. The first part is a lace expansion analysis of a model called loop-weighted walk. Loop-weighted walk is a non-Markovian model of random walks that are discouraged (or encouraged), depending on the value of a parameter, from completing loops. The model arises naturally as a random walk representation of correlations in a statistical mechanics model called the cycle gas. A challenging aspect of this model is that it is not repulsive, meaning the weight of the future of a walk may either increase or decrease if the past is forgotten. Loop-weighted walk is the first finite range walk model with this property to be analyzed with lace expansion techniques.
The second part of this thesis is an essentially elementary derivation of a random walk representation for the partition function of the Ising model on any finite graph. Such representations have a long history for planar graphs. For non-planar graphs the additional ingredient needed is a way to compute the intersection numbers of curves on surfaces. The representations for non-planar graphs lead to random walk representations of spin-spin correlation functions that were previously unknown.
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Case Western
Wed 3 Dec 2014, 3:00pm
One Time Event
ESB 4127 (host: UAlberta)
TBA
ESB 4127 (host: UAlberta)
Wed 3 Dec 2014, 3:00pm-4:00pm

Details

 
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Sat 10 Jan 2015, 9:00am SPECIAL
One Time Event
To Be Announced
Analysis - Qualifying Exams
To Be Announced
Sat 10 Jan 2015, 9:00am-12:00pm
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Sat 10 Jan 2015, 1:00pm SPECIAL
One Time Event
To Be Announced
Differential Equations - Qualifying Exams
To Be Announced
Sat 10 Jan 2015, 1:00pm-4:00pm
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Sat 10 Jan 2015, 1:00pm SPECIAL
One Time Event
To Be Announced Later
Algebra Qualifying Exams
To Be Announced Later
Sat 10 Jan 2015, 1:00pm-4:00pm
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