It's alive!

Today I finally finish proving the result we wanted for my PhD thesis. It has been an epic journey. I wanted to find a rate function for a large deviation principle of the density in the first finite number of sites of a semi infinite totally asymmetric simple exclusion process.

I think that unless one is familiarised with the subject my topic is full of technical vocabulary. So I'll be brief on the explanation of what these all mean.

On one hand there's the large deviation principle. Say you have a sequence of random objects, say tossing coins $X_n$ where they have the value 1 if it's heads and 0 if it's tails, and consider the average of the first $n$ tosses. \[S_n=\frac{1}{n}\sum_{k=1}^nX_k\]

By the law of large numbers we know that this average converges to the actual probability of the toss showing heads, say $p$. Mathematically, for all $\varepsilon>0$ we have that
\[\lim_{n\rightarrow\infty}\mathbb P[|S_n-p|\leq\varepsilon]=1\]

Say we don't want to find the probability of $S_n$ being close to $p$ but any other number $x$, then that limit would be zero:
\[\lim_{n\rightarrow\infty}\mathbb P[|S_n-x|\leq\varepsilon]=0\]

What if we don't take the limit? What if we just take a big natural number $n$? The probability would be close to 0, but positive. Maybe, we can write it like this:
\[\mathbb P[|S_n-x|\leq\varepsilon]\approx\exp\{-nI(x)\}\]

Theory has been developed to explicitly find the rate function. The coin tossing example is the simplest one and can actually be done by hand. However, in general this is not an easy task. The rate function allows you to now the exponential decay rate of the probabilities of rare events, so it's relatively easy to think of applications where this might come in handy.

I have some slides on this here. I think I'll explain what the TASEP is in another post, but the point here is that I proved an LDP for that and I just need to write it down to have a thesis... now I need to finish and get a new job.