Week
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Dates
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Daily
schedule |
Homework
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1
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Aug 27-31
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Mon: no
class
Wed: first day of class. The plan for the first
few lectures is to cover sections 2.1-2.6 from the
Lyons-Peres book which you can download
here.
We covered discrete differential operators and
harmonic functions.
Fri: We went over the Gambler's ruin problem,
introduced electrical networks and proved a
formula relating the strength of the current to
the probability that random walk hits Z before
returning to a.
Register
to vote! (The deadline is Oct 9)
Voting info (written for UT students)
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Homework #1
From the Lyons-Peres book: 2.5, 2.8,
2.13, 2.18, 2.61, 2.65, 2.67.
Due: Mon
Sept 10.
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2
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Sept
3-7
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Mon:
Labor Day (no class).
Tues: last day graduate
students may register and pay tuition without
the approval of the graduate dean.
Wed: The plan for the week is to continue with
chapter 2 (random walks and electrical networks).
We discussed Green's function, current and random
walks.
Fri: We discussed energy, decomposing l^2(E),
current minimizes energy. (This is all in 2.4 of
Lyons-Peres)
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3
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Sept
10-14
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Mon: The plan for
the next 2 lectures at least: series law, parallel
law, Rayleigh's monotonicity theorem,
Nash-Williams, Polya' Theorem, rough embeddings,
transience/recurrence.
Wed:
Fri: We finished 2.5 and covered 2.6 (quickly). We
won't go over the rest of chapter 2 (it is
interesting but not directly related to future
developments).
Last day to drop a class for a possible refund.
Last day a graduate
student may, with the required approvals, add a
class
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Homework #2
From the Lyons-Peres book: 2.54, 2.71,
2.104.
Due: Mon
Sept 17.
Also: if you're reading along, we'll start on
chapter 3 next week.
Solutions
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4
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Sept 17-21
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Mon: we discussed
flows on directed networks, admissible flows,
branching number, its relation to growth rate, and
started on the Max Flow-Min Cut Theorem.
Wed: The plan is to continue with Max Flow-Min Cut
and from there on to homesick random walk on
trees.
Fri: |
Homework #3
From the Lyons-Peres book: 3.4, 3.5,
3.16.
Due: Wed
Sept 26.
Also: if you're reading along, we'll probably
start on chapter 4 next week. |
5
|
Sept
24-28
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Mon: Prof. Baccelli
will give a guest lecture (on point processes and
continuum percolation).
Wed:
Fri: We started Chapter 4 by going over Wilson's
Algorithm.
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6
|
Oct
1-5
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Mon: The plan is to
continue with Chapter 4 (uniform spanning trees).
We might finish in one day because we're not going
over 4.3.
As needed, we'll set up individual meetings for
choosing projects.
Wed: We started on percolation theory, a little of
5.2 and 7.4.
Fri: We covered 5.2 and a little of 5.3
(percolation on trees).
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Homework #4
From the Lyons-Peres book: 4.23, 4.29,
4.30.
Due: Mon
Oct 8.
Also: if you're reading along, we'll probably
start on chapter 4 next week. |
7
|
Oct
8-12
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Mon: The plan is to
finish 5.3 (p_c = 1/br for trees) and start
chapter 6 (isoperimetry). The subject starts with
amenability of graphs. Sometime later in the week,
I plan to lecture on amenability for groups.
Wed: We covered amenability of groups.
Fri: We started on 6.1 (amenability of graphs).
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Homework #5
From the Lyons-Peres book: 5.38, 6.1,
6.3.
Due: Mon
Oct 15.
Also: if you're reading along, we'll probably
start on chapter 7 next week. |
8
|
Oct
15-19
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Mon: We started on
spectral radius and asymptotic behavior of
transition probabilities.
Wed: We went over spectral radius.
Fri: (soft) deadline for choosing a project. We
discussed ergodicity and the number of infinite
components of percolation. We barely started
introducing unimodular networks. The paper
introducing unimodular networks is here.
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Homework #6
From the Lyons-Peres book: 6.19, 7.4,
7.10
Also: prove that the d-regular tree, thought of as
a unimodular network (rooted anywhere) is a limit
of unimodular networks supported on finite graphs.
Due: Mon
Oct 22.
Next week we'll be going over select sections of
chapter 7 and unimodular
networks. |
9
|
Oct
22-26
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Mon: The plan is to
continue with unimodular networks and the number
of ends of infinite components in Bernoulli
percolation.
Wed: We discussed graphings and unimodular
networks to show that the infinite percolation
cluster rooted at the origin of a Cayley graph is
a unimodular random rooted graph.
Fri: We discussed how the # of ends in a
unimodular random graph.
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Hwk
#7 (click to open a pdf file) |
10
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Oct 29
-
Nov 2
|
Mon: No office hours
in the morning. (Instead I'm holding office hours
for a different class). I'll still have office
hours after class as usual. The plan is to go over
amenability of unimodular random graphs.
Wed: We showed that
amenable unimodular random networks have at most
2 ends a.s.
Fri: We started on Gibbs random fields and
showed that Gibbs and Markov random fields are the
same (on finite graphs).
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no homework due next week.
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11
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Nov
5-9
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Mon: The plan is to
introduce the Ising, Potts and random-cluster
models using Grimmett's books and Pete's notes
(from online
resources).
Tues: Voting Day!
Wed: The plan to continue with the random-cluster
model on Z^d and on trees.
Fri: I explained why there is a nontrivial phase
transition in the random cluster model on Z^d for
d>1.
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Homework #8
From Grimmett's Probability
on Graphs: 8.2, 8.8, 8.9.
Due: Mon
Nov 12. |
12
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Nov
12-16
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Mon: I'll sketch
some of the latest developments on the random
cluster model in Z^2 (mostly by sketching the
Harris-Kesten Theorem about percolation) and start
on the Ising model on trees by way of the Evans-Kenyon-Peres-Schulman
paper "Broadcasting on trees and the Ising
model"
Wed: Probably, we'll still be on the Ising model
on trees.
Fri: I think that by Friday, we'll be starting
with Gibbs measures on random sparse graphs ala Amir
Dembo and Andrea Montanari. We might start
with Wormald's
survey on random regular graphs.
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13
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Nov
19-23
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Mon:
Wed: Thanksgiving (no class)
Fri: Thanksgiving (no class) |
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14
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Nov
26-30
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Mon: Last day a graduate
student may change registration in a class to or
from the credit/no credit basis.
Wed:
Fri: |
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15
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Dec
3-7
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Mon:
Wed:
Fri:Last day to submit
master’s report, recital, thesis, doctoral
dissertation, or treatise to the graduate dean |
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16
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Dec
10-14
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Mon: Last class day.
Also last day a graduate student may, with the
required approvals, drop a course or withdraw from
the University.
Wed:
Fri:
Sat: Our final exam is officially on Saturday
2-5pm.
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