# A formal definition of Scaling Limits?

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.

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Can you give us some context? Are you interested in say, Conformal Invariance? –  Alex R. Sep 5 '11 at 4:52
Yes I'm interested in Conformal Invatiance, I'm studying the papers of Werner, Schramm, Lawler,...but intuitively I understand what is scaling limit, but I really wish a more formal definition. My best shot is this: For a proccess which takes values in a lattice $L\subseteq \mathbb{Z}^d$, the limit of the proccess $\delta L$ as $\delta$ tends to zero. Is needed to be a graph in the reals? –  Murphy Sep 5 '11 at 14:03
You might also want a definition for "processes" which have a lattice as a domain instead of the set where it takes its values. This is the situation when approximating a quantum field theory using lattice theories. The best reference for this is probably section 3 of the Seminaire Bourbaki by Frohlich and Spencer: numdam.org/numdam-bin/fitem?id=SB_1981-1982__24__159_0 –  Abdelmalek Abdesselam Sep 5 '11 at 19:38

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 $\phi_n(\xi_n)$.

1. One point of view is understanding scaling limits as limiting points for renormalization group.

2. Papers by P.Major from around 1980 on self-semilarity and renormalization are useful in understanding the concept.

3. 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).

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I think the main issue you'll encounter with scaling limits and conformal invariance will be that it is a very new subject. Many of the papers will be very high level or still in preprint with details missing.

Here's a list of references that helped me learn the subject. I'll try to comment on some of them later on when time permits. I would first start with "Scaling Limits and SLE" by Greg Lawler. This set of notes is in context of self avoiding walks and how they connect to SLE and you'll find they quickly dive into scale invariance and conformal invariance.

If you are further interested in proofs, I would consult Lawler's "Conformally Invariant Processes in the Plane." This book gives very rigorous formulations of SLE and tackles a myriad of technical difficulties.

Next I would look at "Toward conformal invariance of 2D lattice models" by Smirnov. Here you'll be introduced to the notion of the duality between holomorphic martingales and scale invariance. In particular, you need to understand how discrete complex analysis connects with conformal invariance. I would hold out for proofs of many of the results for now.

Now you'll probably want to look at some simpler examples. I would first start trying to understand the proof of Cardy's Formula. Geoffrey Grimmett has an excellent set of lecture notes, "Probability on Graphs" which should be available off Grimmett's website (I can't seem to access it right now). As well I would consult the original paper by Smirnov, "Critical percolation in the plane". I'll add here that the proof of Cardy's formula doesn't need the full machinery of holomorphic martingales because of the really nice symmetries Smirnov observed. What IS important though is how the Riemann Hilbert boundary conditions are set up.

One of the big issues with discrete complex analysis is how to rigorously define discrete holomorphic functions on graphs. If you want the gory details, "Discrete complex analysis on isoradial graphs" by Chelkak and Smirnov is a good place to look.

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I think the notion of scaling limit is really more of a group of ideas than a single definition, since there are different types of objects being studied and hence different notions of convergence. Typically though one considers convergence of some (rescaled) sequence of probability measures on "discrete" objects as a spatial parameter $\delta\rightarrow0$ to a probability measure on some "continuum" object. Part of the trouble is that it can be tricky to figure out what the right "continuum" object is and further how to put a probability measure on it. A nice discussion about scaling limits and conformal invariance are these lecture notes by Lawler.

The fundamental example of such a scaling limit is the convergence of random walks to Brownian motion; one source is chapter 5 of the book of Mörtens and Peres.

An interesting scaling limit with a different flavor than the SLE ones described in other answers is Aldous's Continuum Random Tree. The following is from the linked page:

Take a critical Galton-Watson branching process where the offspring law has finite non-zero variance, and condition on total population until extinction being $n$. This gives a random tree. Rescale edge-lengths to have length $n^{-1/2}$. Put mass $1/n$ on each vertex. In a certain sense that can be formalized, the $n \to \infty$ weak limit of these random trees is the Brownian CRT (up to a scaling factor).

This is a beautiful object on its own but also a key tool in several other scaling limits.

The CRT is used in the study of random planar maps, see these lecture notes of Le Gall and Miermont. Recent developments not covered in these notes are their proofs of the convergence of certain families of random planar maps to the so-called Brownian map (see here and here).

The CRT is also part of the construction of the scaling limit of connected components of Erdős–Rényi random graphs in the scaling window.

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I was listening to Lawler give a series of talks this summer and his sense of scaling limit is the same as is often used in defining Brownian motion as a scaling limit of simple random walks. In which both time and space have to be scaled so that the limit process doesn't become trapped at the origin. The difference between Brownian motion and SLE is that the random walks are no longer simple. The conformal invariance in SLE comes of the properties of the not-simple random walks if memory serves.

So if one wanted a rigorous definition of a scaling limit I would look for a text on Brownian motion (most any probability with measure theory book will do this construction in the continuous time chapter) or a more specialized book on continuous time processes. There should be plenty of these in most any language you'd want.

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