I'm looking for nice overviews on $\phi^{4}$-field theory from the mathematical-physics point of view. To be a little more specific, here are some topics I'd like to read about:


**(1)** What are the motivations, both from the physics and mathematical point of view, to study $\phi^{4}$-theories?


**(2)** What has been (mathematically) accomplished so far? What are the most important open questions nowadays?


**(3)** What are the tools used to (rigorously) study these class of models? Renormalization group, cluster expansions, etc.


My motivation for this question is a rather simple one: I'm teaching myself some field theory but it is really hard to find these discussions on books. In general, books are more interested in solving problems and sometimes I find myself studying some models that I don't know anything about. Also, I'm primarily interested in statistical mechanics, so this helps to narrow the question a bit more. Connections with statistical mechanics and QFT are welcome too, but I don't want a purely QFT reference (I'm sufficiently lost in my own area of interest, after all). Thanks in advance!



Good overviews on $\phi^{4}$-field theory?
This reference is a bit older, but it should be a good starting point for items 2 and 3: [$\phi^4$ field theory in dimension 4: a modern introduction to its unsolved problems](http://ipparco.roma1.infn.it/pagine/deposito/1967-1979/086.pdf).


Concerning item 1, you might find it instructive to motivate the $\phi^4$ field theory from the perspective of its limitation: when does it apply and when are higher order terms needed? For that perspective I would recommend [Higher-order field theories: $\phi^6$, $\phi^8$ and beyond.](https://arxiv.org/abs/1806.06693)


This last reference is a chapter from a recent book, [A Dynamical Perspective on the $\phi^4$ Model](https://www.springer.com/gp/book/9783030118389) (2019) which has an interesting introductory chapter on the history of the $\phi^4$ model, as well as overviews of more specialized topics.


