TITLE: A formal definition of Scaling Limits?
QUESTION [5 upvotes]: I'm looking for a formal definition of scaling limit in a rigorous math sense, also, if somebody knows a good translation to spanish. A good bibliography could be helpful.

REPLY [8 votes]: This is the idea. Suppose that you have a family or sequence of structures of growing complexity (long random walks, realizations of a random field on a large piece of a lattice or in a large continuous domain, large random trees, etc.). You want to understand the behavior of the large structures of your family. Often you want to say that your large random object is similar to a simpler object that you can describe precisely. Since the random walk consisting of 1000 steps is quite different from the "same" random walk consisting of 100000 steps, but you still want to find similarities between them, it makes sense to normalize or rescale your objects appropriately. If you manage to find the right rescaling (it is given by shrinking the time by $n$ and space by $n^{1/2}$ for a standard simple symmetric random walk), then you might discover that thus rescaled (and appropriately embedded into the space of continuous functions or the Skorokhod space) random walk converges in distribution to the Wiener process.
So, scaling limits provide approximative descriptions of what your objects look like when "you look at them from a large distance" or "zoom out".
At a more formal level, suppose $\xi_n$ is a sequence of random objects in some space $X$ . Suppose $\phi_n$ is a (carefully chosen) sequence of scaling tansformations in $X$. It is hard to say precisely what a scaling transformation is, often it is a linear map depending on $n$ with coefficients decaying in $n$. Often, a (time)-reparametrization of the random objects involved is a part of $\phi_n$. A scaling limit is the distributional limit of $\phi_n(\xi_n)$.
Some more comments:

One point of view is understanding scaling limits as limiting points for renormalization group.
Papers by P.Major from around 1980 on self-semilarity and renormalization are useful in understanding the concept.
I will take this chance to advertise my own paper on scaling limits for random trees, where I describe what large random trees look like if drawn on the plane and looked at from a large distance. It appeared this year in Markov Processes and Related Fields and is also available at http://arxiv.org/abs/0909.2283  The construction and scaling used is different from Aldous's continuum trees (and there are strong connections to superprocesses).