There are several interesting properties that become visible in large random systems like large random graphs, or random walks or random matrices etc...

For example, in such systems one observes:

- Averaging and concentration: the simplest example is the Law of Large numbers, where the average of i.i.d. random variables converges to its mean. In fact, more generally functions of many independent random variables will often be close to their expectation and one can quantitatively bound the flucatuations.

- Universality: in different large systems microscopic properties might lose their importance and some universal properties appear. The simplest example is the Central limit theorem - if you add up i.i.d. random variables with finite variance and normalize properly, the Gaussian law always appears and the fine details of initial laws of the random variables don't matter. There are much richer examples of such phenomena.

- Phase transitions: functions of several independent random variables often change their behaviour very abruptly when changing a certain parameter. For example think of the sum of n i.i.d. random variables with values +1 or -1: if the situation is symmetric, this sum is of order O(\sqrt{n}, if P(X = 1) > P(X = -1), the sum will behave like f(n) = cn and otherwise like f(n) = -cn, for some positive constant c.

We will try to look into some of these topics and the methods used to study them.