Complex systems are a largely unexplored area of physics in that we don't really have tools as powerful and generally applicable as the mathematical machinery developed to describe more "fundamental" physics. With the exception of near equilibrium phenomena of homogeneous systems like gases that can be described with 19th century statistical physics it is really hard to write down general laws or patterns.
The work of Parisi (and quite a few others) in the eighties (non-linear systems, chaos theory, attractors, universality of power laws etc) have given us a first glimpse of what lies beyond, but a true revolution is still in the future and will require some pretty mind-bending mathematical inventions.
Making serious progress is not just intellectually challenging, it is also of immense practical relevance for us understanding and moderating our impact on the biosphere. The Nobel committee, in their infinite wisdom, suggest as much.
Parisi work on spin glasses already has some pretty mind bending mathematics. That continuous limit where the rank of a matrix goes to zero for example?
+1 on the whole references as a nice introduction. I think the authors overstate the preparation of their hypothetical "pedestrian" (either that or they need to get away from the physics department a bit more often), but a great reference nevertheless. I also got a lot out of sections of Nishimori's textbook [1]. In particular it helps motivate problems outside of physics and provides some references to start digging into more rigorous approaches via cavity methods (which I think, incidentally, are also more intuitive). I am a novice in this area but am sort of crossing my fingers that some of the ideas in this area will make their way into algorithms for inferring latent variables in some of the upcoming modern associative neural networks. What I mean here is that it would be cool not just to have an understanding of total capacity of the network but also correct variational approximations to use during training.
Let me take a stab at this (I'll maybe take it halfway there). First of all we want to know what kind of matrix we are talking about.
Imagine that you have a whole bunch generative models (its best if you imagine a fully connected Boltzmann machine in particular, whose states you can think of as a binary vector consisting only of zeros and ones) that have the same form but different random realizations of their parameters. This is a typical example of what a toy model of a so-called "spin glass" looks like in statistical physics (the spins are either up down down, usually represented as +1/-1). Each of these models, having been initialized randomly will have their up particular frequency of a particular location (also called site) of the boolean vector being either a one or a zero.
If the tendency of a site to be either or one or a zero was independent of every other site the analysis of such a model would be pretty straightforward: every model would just have a vector of N frequencies and we could compare how close the statistical behavior of each model was to the other by comparing how closely the N frequencies at each site matched one another. But in the general case there will be some interaction or correlation between sites in a given model. If the interaction strength is strong enough this can result in a model tending to generate groups of patterns in its sequence of zeros and ones that are close to one another. Furthermore if we compare the overlap of the apparent patterns between two such models, each with their own random parameters, we will find that some of them overlap more than others.
What we can then do is to ask the question of how much, on average do the patterns of these random models overlap with on another in the full set of all models. This leads us to the concept of an "overlap matrix". This matrix will have one set of values along the diagonal (corresponding to how much a models patterns tend to overlap with themselves) and off diagonal values capturing the overlap between. You can find through simulation or with some carefully constructed calculations that when the interaction strength between sites is small that the off diagonal elements don't tend to zero, but rather a single number different from the diagonal value. This is perhaps intuitive: these models were randomly initialized but they are going to overlap in their behavior in some places.
Where things get interesting though is when you increase the interaction strength you find that the overlap matrix starts to take on a block diagonal form, wherein clusters of models overlap with one another at a certain level and at a lower but constant level with out-of-cluster models. This is called one replica symmetry breaking (1RSB). These different clusters of models can be thought of as having learned different overall patterns with the similarity quantified by their overalp. If you keep increasing the interaction strength you will find that this happens again and again, with a k-fold replica symmetry braking (kRSB) with a sort of self similar block structure emerging in the overlap matrix (picture is worth a thousand words [1]).
Now the real wild part that Parisi figured out is what happens when you take this process to the regime of full replica symmetry breaking. You can't really do this with simulations and the calculations are very tricky (you have a bunch of terms either going to infinity or zero that need to balance out correctly) but Parisi ending up coming up with an expression for the distribution of overlaps for the infinitely sized matrix with full interaction strength in play. The expression is actually a partial differential equation that itself needs to be solved (I told you the calculations were tricky right), but amazingly, it seems to capture the behavior of these kinds of models correctly.
Whereas mathematicians have a pretty good idea of how to understand the 1RSB process rigorously, the Parisi full replica symmetry breaking scheme is very much not understood and remains of interest both to complex systems researches trying to understand their models and applied mathematicians (probability people in particular) trying to lay the foundations needed to explore the ideas being explored by theorists.
> The Nobel committee, in their infinite wisdom, suggest as much.
Usually this phrase is used ironically, as in “they don’t know what the [expletive] they’re doing,” but the rest of your comment reads genuine. Because people seem split on this and because I don’t fully understand any of it, can you clarify one thing: are you throwin’ shade?
The work of Parisi (and quite a few others) in the eighties (non-linear systems, chaos theory, attractors, universality of power laws etc) have given us a first glimpse of what lies beyond, but a true revolution is still in the future and will require some pretty mind-bending mathematical inventions.
Making serious progress is not just intellectually challenging, it is also of immense practical relevance for us understanding and moderating our impact on the biosphere. The Nobel committee, in their infinite wisdom, suggest as much.