Cool Pictures in Probability
Sunil Chhita
September 18, 2008
Sunil Chhita Cool Pictures in Probability
Introduction
Originally used to study gambling models. During the 20th
Century, probability theory was made into a rigorous field.
Sunil Chhita Cool Pictures in Probability
Introduction
Originally used to study gambling models. During the 20th
Century, probability theory was made into a rigorous field.
This talk shall focus on a few models that have a pictorial
representation.
Sunil Chhita Cool Pictures in Probability
Introduction
Originally used to study gambling models. During the 20th
Century, probability theory was made into a rigorous field.
This talk shall focus on a few models that have a pictorial
representation.
We aim to walk the walks, tile the tiles and diffuse the
diffusions.
Sunil Chhita Cool Pictures in Probability
Introduction
Originally used to study gambling models. During the 20th
Century, probability theory was made into a rigorous field.
This talk shall focus on a few models that have a pictorial
representation.
We aim to walk the walks, tile the tiles and diffuse the
diffusions.
But first, a timeline:
Sunil Chhita Cool Pictures in Probability
Simple symmetric Random Walk on a line
Suppose you start at the origin. To decide where to move,
toss a fair coin.
Sunil Chhita Cool Pictures in Probability
Simple symmetric Random Walk on a line
Suppose you start at the origin. To decide where to move,
toss a fair coin.
If you get a heads, move left and if you get a tails move
right.
Sunil Chhita Cool Pictures in Probability
Simple symmetric Random Walk on a line
Suppose you start at the origin. To decide where to move,
toss a fair coin.
If you get a heads, move left and if you get a tails move
right.
Plotting the distance from the origin at each time step gives
Sunil Chhita Cool Pictures in Probability
Some properties of a random walk
The position of the walk at the next step only depends on
the position of the current level.
Sunil Chhita Cool Pictures in Probability
Some properties of a random walk
The position of the walk at the next step only depends on
the position of the current level.
Our random walk is symmetric-the probability of going to
the left is the same as the probability of going to the right.
Sunil Chhita Cool Pictures in Probability
Suppose we allow a walk to operate for a larger time.
Sunil Chhita Cool Pictures in Probability
Suppose we allow a walk to operate for a larger time.
Sunil Chhita Cool Pictures in Probability
Suppose we allow a walk to operate for a larger time.
Scaling this large time random walk in a certain way leads
to Brownian Motion. This type of limit is called a scaling
limit.
Sunil Chhita Cool Pictures in Probability
Brownian Motion on a line
First noticed by the botanist Robert Brown but brought to
light for the physicists by Einstein in 1905.
Sunil Chhita Cool Pictures in Probability
Brownian Motion on a line
First noticed by the botanist Robert Brown but brought to
light for the physicists by Einstein in 1905.
Norbert Wiener proved the existence of the process in
1923.
Sunil Chhita Cool Pictures in Probability
Some Properties of a Brownian Motion
Three main properties of a Brownian Motion, Bt
It’s continuous (almost surely).
Sunil Chhita Cool Pictures in Probability
Some Properties of a Brownian Motion
Three main properties of a Brownian Motion, Bt
It’s continuous (almost surely).
It has independent increments, and Bt − Bs is Gaussian,
mean 0 and variance t − s.
Sunil Chhita Cool Pictures in Probability
Some Properties of a Brownian Motion
Three main properties of a Brownian Motion, Bt
It’s continuous (almost surely).
It has independent increments, and Bt − Bs is Gaussian,
mean 0 and variance t − s.
Starts at the origin.
Sunil Chhita Cool Pictures in Probability
Applications of Brownian Motion
Finance-to model stock prices.
Sunil Chhita Cool Pictures in Probability
Applications of Brownian Motion
Finance-to model stock prices.
Transport phenomena-modelling tiny particles in high
speed gas flows.
Sunil Chhita Cool Pictures in Probability
Applications of Brownian Motion
Finance-to model stock prices.
Transport phenomena-modelling tiny particles in high
speed gas flows.
Decision Analysis-optimal switching times in economic
activity.
Sunil Chhita Cool Pictures in Probability
Applications of Brownian Motion
Finance-to model stock prices.
Transport phenomena-modelling tiny particles in high
speed gas flows.
Decision Analysis-optimal switching times in economic
activity.
Heart of Gold-Calculate the Infinite Improbability Drive.
(The Hitch-hikers guide to the galaxy.
Sunil Chhita Cool Pictures in Probability
Some other properties of Brownian Motion
In one dimension, it is recurrent.
Sunil Chhita Cool Pictures in Probability
Some other properties of Brownian Motion
In one dimension, it is recurrent.
It is not differentiable.
Sunil Chhita Cool Pictures in Probability
Some other properties of Brownian Motion
In one dimension, it is recurrent.
It is not differentiable.
It is scale invariant.
Sunil Chhita Cool Pictures in Probability
Some other properties of Brownian Motion
In one dimension, it is recurrent.
It is not differentiable.
It is scale invariant.
It is a martingale.
Sunil Chhita Cool Pictures in Probability
Some other properties of Brownian Motion
In one dimension, it is recurrent.
It is not differentiable.
It is scale invariant.
It is a martingale.
It satisfies the Markov Property and Strong Markov
Property.
Sunil Chhita Cool Pictures in Probability
Mathematical applications of BM.
Stochastic Analysis.
Sunil Chhita Cool Pictures in Probability
Mathematical applications of BM.
Stochastic Analysis.
Interacting Particle Systems-e.g. contact process.
Sunil Chhita Cool Pictures in Probability
Mathematical applications of BM.
Stochastic Analysis.
Interacting Particle Systems-e.g. contact process.
Potential Theory.
Sunil Chhita Cool Pictures in Probability
Mathematical applications of BM.
Stochastic Analysis.
Interacting Particle Systems-e.g. contact process.
Potential Theory.
Analysis.
Sunil Chhita Cool Pictures in Probability
Brownian Motion in 2-d
Sunil Chhita Cool Pictures in Probability
Domino Tilings
Want to randomly tile a shape on a plane with dominos
(NOT THE PIZZA).
Sunil Chhita Cool Pictures in Probability
Domino Tilings
Want to randomly tile a shape on a plane with dominos
(NOT THE PIZZA).
A shape consists of a connected collection of unit squares.
Sunil Chhita Cool Pictures in Probability
Domino Tilings
Want to randomly tile a shape on a plane with dominos
(NOT THE PIZZA).
A shape consists of a connected collection of unit squares.
A domino covers two adjacent squares. A domino tiling
refers to covering the shape with dominos so that every
square in the shape is covered.
Sunil Chhita Cool Pictures in Probability
Domino Tilings
Want to randomly tile a shape on a plane with dominos
(NOT THE PIZZA).
A shape consists of a connected collection of unit squares.
A domino covers two adjacent squares. A domino tiling
refers to covering the shape with dominos so that every
square in the shape is covered.
E.g. For a checkerboard
Sunil Chhita Cool Pictures in Probability
Question?
Suppose that we remove two opposite corner squares from
the checkerboard.
Sunil Chhita Cool Pictures in Probability
Question?
Suppose that we remove two opposite corner squares from
the checkerboard.
Can this shape be tiled with dominos?
Sunil Chhita Cool Pictures in Probability
Other Regions
We can count the number of tilings of a chessboard of
arbitrary size.(For an 8 by 8 board there’s 12988816
tilings!)(This can be found via Kastelyn’s Theorem)
Sunil Chhita Cool Pictures in Probability
Other Regions
We can count the number of tilings of a chessboard of
arbitrary size.(For an 8 by 8 board there’s 12988816
tilings!)(This can be found via Kastelyn’s Theorem)
We can pick any of these tilings at random to generate a
random tiling.
Sunil Chhita Cool Pictures in Probability
Other Regions
We can count the number of tilings of a chessboard of
arbitrary size.(For an 8 by 8 board there’s 12988816
tilings!)(This can be found via Kastelyn’s Theorem)
We can pick any of these tilings at random to generate a
random tiling.
Tiling randomly a square or a rectangle is boring. Instead
let’s focus on an Aztec Diamond.
Sunil Chhita Cool Pictures in Probability
An Aztec Diamond is a shape of the following form:
Sunil Chhita Cool Pictures in Probability
Tiling a large Aztec Diamond randomly leads to
Sunil Chhita Cool Pictures in Probability
More impresively
This is called the Arctic Circle phenomenon, whereby taking a
tiling gives a limit shape.
Sunil Chhita Cool Pictures in Probability
Types of dominos
Four ways to tile a domino: two vertical and two horizontal.
This can be observed from the chessboard.
Sunil Chhita Cool Pictures in Probability
Types of dominos
Four ways to tile a domino: two vertical and two horizontal.
This can be observed from the chessboard.
To simplify the diagram, place a dot at the corner where
below there is a black square to the left
Sunil Chhita Cool Pictures in Probability
Types of dominos 2
Our diamond now looks like this
Sunil Chhita Cool Pictures in Probability
Type of dominos 3
To distinguish between the two vertical (and two horizontal)
dominos, label as follows
Sunil Chhita Cool Pictures in Probability
Type of dominos 3
To distinguish between the two vertical (and two horizontal)
dominos, label as follows
For a 2 by 2 block of dominos, if two arrows meet label the
block BAD otherwise the block is GOOD.
Sunil Chhita Cool Pictures in Probability
Steps of Shuffling Algorithm
The shuffling algorithm generates a tiling of an Aztec
diamond of order n + 1 from a tiling of an Aztec diamond of
order n in three simple steps:
Sunil Chhita Cool Pictures in Probability
Steps of Shuffling Algorithm
The shuffling algorithm generates a tiling of an Aztec
diamond of order n + 1 from a tiling of an Aztec diamond of
order n in three simple steps:
Step 1: Destruction, remove the bad blocks
Sunil Chhita Cool Pictures in Probability
Steps of Shuffling Algorithm
The shuffling algorithm generates a tiling of an Aztec
diamond of order n + 1 from a tiling of an Aztec diamond of
order n in three simple steps:
Step 1: Destruction, remove the bad blocks
Step 2: Sliding, move all the remaining blocks in the
directions of their arrows
Sunil Chhita Cool Pictures in Probability
Steps of Shuffling Algorithm
The shuffling algorithm generates a tiling of an Aztec
diamond of order n + 1 from a tiling of an Aztec diamond of
order n in three simple steps:
Step 1: Destruction, remove the bad blocks
Step 2: Sliding, move all the remaining blocks in the
directions of their arrows
Step 3: Creation, fill in with good 2 by 2 blocks, vertical
1
with a probability of 1 and horizontal with probability of 2 .
2
Sunil Chhita Cool Pictures in Probability
Example of Shuffling Algorithm
Sunil Chhita Cool Pictures in Probability
This limit shape is not unique to dominos on an Aztec Diamond.
For example, lozenges (not cough medication) tiled on the Box
Plane Partition gives
(Picture by Rick Kenyon)
Sunil Chhita Cool Pictures in Probability
Other pictures-Random Matchings
Suppose we have points consisting of two colours, red and
blue, randomly arranged in a space (2 dimensions).
Sunil Chhita Cool Pictures in Probability
Other pictures-Random Matchings
Suppose we have points consisting of two colours, red and
blue, randomly arranged in a space (2 dimensions).
Suppose the number of blue points is equal to the number
of red points.
Sunil Chhita Cool Pictures in Probability
Other pictures-Random Matchings
Suppose we have points consisting of two colours, red and
blue, randomly arranged in a space (2 dimensions).
Suppose the number of blue points is equal to the number
of red points.
A matching corresponds to linking a blue point with a red
point.
Sunil Chhita Cool Pictures in Probability
Other pictures-Random Matchings
Suppose we have points consisting of two colours, red and
blue, randomly arranged in a space (2 dimensions).
Suppose the number of blue points is equal to the number
of red points.
A matching corresponds to linking a blue point with a red
point.
Have a perfect matching if all the points are matched.
Sunil Chhita Cool Pictures in Probability
Other pictures-Random Matchings
Suppose we have points consisting of two colours, red and
blue, randomly arranged in a space (2 dimensions).
Suppose the number of blue points is equal to the number
of red points.
A matching corresponds to linking a blue point with a red
point.
Have a perfect matching if all the points are matched.
Can match up the points so that the distance between all
points in minimum.
Sunil Chhita Cool Pictures in Probability
(Picture by Ander Holroyd)
Sunil Chhita Cool Pictures in Probability
Or can match them up so they are ‘stable’ (using a the greed
algorithm).
(Picture by Ander Holroyd)
Sunil Chhita Cool Pictures in Probability
Other Pictures-Stable Marriage
(Picture by Ander Holroyd)
Sunil Chhita Cool Pictures in Probability
Think of the ‘dots’ as centres and the other points as sites.
The centres are located randomly.
Sunil Chhita Cool Pictures in Probability
Think of the ‘dots’ as centres and the other points as sites.
The centres are located randomly.
Each centre wants to claim territory, represented by its
circle.
Sunil Chhita Cool Pictures in Probability
Think of the ‘dots’ as centres and the other points as sites.
The centres are located randomly.
Each centre wants to claim territory, represented by its
circle.
Each territory needs to be the same size.
Sunil Chhita Cool Pictures in Probability
Think of the ‘dots’ as centres and the other points as sites.
The centres are located randomly.
Each centre wants to claim territory, represented by its
circle.
Each territory needs to be the same size.
Sites prefer a nearby centre whilst centres prefer a nearby
territory.
Sunil Chhita Cool Pictures in Probability
Think of the ‘dots’ as centres and the other points as sites.
The centres are located randomly.
Each centre wants to claim territory, represented by its
circle.
Each territory needs to be the same size.
Sites prefer a nearby centre whilst centres prefer a nearby
territory.
Unstable if there is a site and a centre which are not
allocated to each other, but both prefer each other
compared to its current partition.
Sunil Chhita Cool Pictures in Probability
Other Pictures-Uniform Sorting Networks
(Picture by Ander Holroyd)
Sunil Chhita Cool Pictures in Probability
Other Pictures-Uniform Sorting Network.
(Picture by Ander Holroyd)
Sunil Chhita Cool Pictures in Probability
Rotor Model
(Picture by Yuval Peres)
End of Talk. Thanks for listening.
Sunil Chhita Cool Pictures in Probability