11 milli-blog - math musings - uncertainty principles and phase transitions

#update
parents:: milli-blog
daily note:: 2025-01-10

This is an agglomeration of a bunch of different observations that I still haven't fully figured out how to interpret.
(This is a very rough post that's highly incomprehensible).

Uncertainty principles:
- Uncertainty principle is most well known in the quantum mechanics setting. Where, for instance, the uncertainty principle between position and momentum says that the product of the variances of the two is lower bounded by some universal constant.
- There are also uncertainty principle in statistical mechanics from the perspective of parameter recovery: e.g. the optimal accuracy in recovering $\beta$ (the inverse temperature) is inversely related to the variance in $E$ (the energy)
  - More generally, any two conjugate variables will satisfy an uncertainty relation
- Outside of the physics context, there are certain uncertainty principles relating the zeroes of a polynomial with its coefficients
  - In fourier analysis on the boolean hypercube, the number of non-zero coefficients of $f$ times the number of non-zero values of $f$ (when evaluated on ${-1,1}^n$ is lower bounded by $2^n$.
    - If you look at this in one direction, this says that if the zeros of $f$ are highly structured, then the $f$ itself must be highly "random" in that it's composed of many different frequencies
  - In Central limit theorems and the geometry of polynomials, the authors show that whenever you have a degree $n$ univariate polynomial whose roots are all bounded away from $1$, then the coefficients of $f$ must be highly "random" in that it looks like a Gaussian (the maximum entropy distribution for a fixed variance)
Phase transitions:
- A phase transition is term that loosely means when some macroscopic observable of a system jumps discontinuously (or behave non-smoothly) at/near a particular value of a parameter that you can vary. For instance in the Ising model, the first derivative of average magnetization vs temperature behave discontinuously at the critical temperature.
- Oftentimes, these observables we care about are derivatives of the log partition function w.r.t certain parameters (let's call it $\lambda$) (such as temperature, or external field in the case of magnetization). So that in particular, since the partition function itself is continuous w.r.t. the parameter $\lambda$, as long as the partition function remains bounded away from 0, these derivatives of the log partition function will remain analytic w.r.t $\lambda$
  - That is, structure of the zeroes tells you something about phase transitions
How they relate
- Another sign of a phase transition is that concentration of measure of some observable fails. For example, in the ferromagnetic ising model, below the critical temperature and with zero external field, the average magnetization is bimodal between $\pm m$ for some value $m$.
  - This is related to the non-uniqueness threshold.
- This is related to the Central limit theorems and the geometry of polynomials since...
  - You can get a univariate polynomial from the ising model partition function whose coefficient of $x^k$ is the probability of getting a state in ${0,1}^n$ whose sum is $k$
  - As long as the zeroes are all far from 1, then the result in the paper says that you will get concentration of measure
  - Do we know that all the zeroes are far from 1? I'm not sure actually... but I would guess so.
- In terms of uncertainty principles, this is sort of like saying that... I'm not quite sure where I'm going here
thermodynamics is to statistical mechanics as classical mechanics is to quantum mechanics
- Both arise in the limit as uncertainty goes to 0

Questions

Is there a version of Central limit theorems and the geometry of polynomials for the wigner's semicircle law?
- wigner's semicircle law is the noncommutative analogue of the gaussian.