The talk explores the concept of large deviations in probability, using the example of coin tosses to illustrate how random variables behave and how averages stabilize as the number of trials increases. It emphasizes the significance of the central limit theorem and the law of large numbers in determining the distributions of outcomes. The research aims to model complex systems through the lens of particle interaction, with applications in fields like insurance and finance.

1. Horacio González Duhart
hgd20@bath.ac.uk

2. The point of this talk is to know what are…
Large
Deviations
Big
Gargantuan
Enormous
Tremendous
Immense
Alterations
Anomalies
Variations
Diversions

3. But first we need to prepare the path… that will lead us to the real of probability…
Probability
=
Las Vegas!

4. Key concept: Random variable
A random variable is (loosely speaking) the result of an experiment that we cannot
predict.
Favourite random experiment for probabilists: tossing a coin.
To keep things simple, we say can say that “heads” is 1 and “not heads” is 0, and the
probability of getting heads is p.
Result
Heads
Not heads
Probability
p
1-p
Coding
1
0

5. One coin? Seriously, one coin?... That’s right, you guessed correctly... Probabilists (as
other common human beings) prefer loads of coins, rather than just one…
So that we don’t get lost, we number our coins from 1 to N, and call the result of
the k-th coin X_k, mathematically:

6. We are interested in studying the average “random behaviour” (which probabilists call
distribution) of throwing all coins.
If we throw 3 coins, we can get 1 of 8
possible results:
Which we encode as vectors of 0s and
1s, for example:
001
110
This simple strategy
has saved probabilists
from using actual
coins for centuries! But we want to look at the average (which
we call mean)
1/3
2/3
Number of heads
Number of coins
Mathematically:
Average of n
tosses
Number of
coins
Sum of the
tosses

7. Let’s analyse this tossing 3
coins experiment closely…
First coin Second coin
Third coin Mean Probability
1
2/3
1/3
0 1/8
3/8
3/8
1/8
The mean seems to
concentrate in the
middle with higher
probability…

8. Let’s literally do the experiment and keep track of how the mean is moving.
First coin
Mean = 0
Second coin
Mean = 1/2
Third coin
Mean = 1/3

9. What about 10 coins?
Here it is…

10. But let’s take it seriously… what about ONE THOUSAND coins?!
Looks like for “large” values of n, the mean is getting more and more stable. Maybe we
were lucky… let’s do this a couple of times more…
Not convinced yet? Let’s do this ONE THOUSAND times!
The distribution of the mean is concentrating as n gets bigger.
Let’s zoom in into what happens when n=10

11. At this point, I want to say that
although the green curve looks
like 0 (at say 0.2) it is always
positive… the thing is that
chances are just too slim now!
So, when n=10
So, when n=100
So, when n=1000
We can even see the
distribution when n=10000
Way out of the plot above!
Some things you already know:
The Central Limit Theorem
states that for large n we will
see this bell-shaped like
distributions.
The Law of Large Numbers
states that these distributions
will be centred around the
“real” probability a coin has to
land on heads.

12. What we want to see is a function that tells us how fast this
concentration is happening
This is the core of object
of study of large
deviations: the rate
function
High values of the rate
function mean we need
relatively few coin tosses
to be unable to produce
this mean.
Low values… we need
many coin tosses to be
unable to produce it.
A value of zero… were we
be able to perform
amount of coin tosses,
we would get this mean.

13. Now that we know large deviations… My research in a nutshell…
1. Put N boxes…
2. Put some balls…
4. Attach and set an alarm clock to each box.
Ball
factory!
5. When an alarm clock rings, if there’s a ball in the box, flip a coin. If heads move the ball
to the right if the box is empty, if not heads move the ball to the left if the box is empty.
3. Start a timer…
If the box is occupied, then there is no movement, only one ball per box allowed!
Yes, balls may fall down from the system if they’re in the last box…
6. Repeat the process from step 4 until the timer hits time T
7. When finished, take the average over space-time of the number of balls.
8. Find a LDP for that average…

14. But… but…

15. Even when most of the times a mathematician would answer some things are
interesting in themselves… it so happens that large deviations are quite useful…
Insurance mathematics and actuarial science…
If there are N insured people, what is the premium p the Insurance company should
charge so that the total claims is less than the total raised income?
From here it’s easy to imagine similar applications in finance (more coins!)

16. Our boxes, balls, clocks example might have seen very artificial… but it isn’t!
It’s just an abstract simplified model of interacting particle systems in statistical mechanics
We could have chosen other examples such as…
A forest on fire and
how the fire is
consuming it…
Bacteria growing on a
Petri dish…
Cars in a traffic jam…
Molecules of water in a
wave….

17. This research may be a small step… but it’s a
beginning to a better understanding of the connection
between the microscopic and macroscopic worlds.

18. So… I created this meme and has been an awesome sensation… among the like 6 maths
geeks following the hashtag #LargeDeviations on Twitter…