Week 2 - Revisiting Parisi's Equations and the Replica Method

Recap of the Meeting

We spent most of our time discussing geometric intuition for the dynamics described by the Parisi’s PDEs.

• The equations are given as $$\{\begin{array}{l}{\mathrm{\Phi }}_t(t,x)=-\frac{{\beta }^2}{2}\cdot ({\mathrm{\Phi }}_{xx}(t,x)+\mu (t)\cdot |{\mathrm{\Phi }}_x(t,x){|}^2)\\ \mathrm{\Phi }(1,x)=\log (2\cdot \cosh x)\end{array}$$

• When a change of variable is performed $t(s)=1-s$, the PDE becomes $$\{\begin{array}{l}{\mathrm{\Phi }}_s(s,x)=\frac{{\beta }^2}{2}\cdot ({\mathrm{\Phi }}_{xx}(s,x)+\mu (1-s)\cdot |{\mathrm{\Phi }}_x(s,x){|}^2)\\ \mathrm{\Phi }(0,x)=\log (2\cdot \cosh x)\end{array}$$

• The behavior of this dynamic can be interpreted as follows. At time $s=0$, we begin with the soft absolute value function $\mathrm{\Phi }(0,x)=\log (2\cdot \cosh x)$. As $s$ progresses from $0\to 1$, the evolution of $\mathrm{\Phi }(s,x)$ is always affected by heat diffusion dynamics $$-\frac{{\beta }^2}{2}\cdot {\mathrm{\Phi }}_{xx}(t,x)$$

• On the other hand, $\mu (1-s)$ is non-increasing on the interval $[0,1]$. The function $\mu (s)$ itself is the cumulative density function of some measure supported on $[0,1]$. Thus, the evolution of $\mathrm{\Phi }(s,x)$ is also affected by a “gradient descent” dynamic $$\frac{{\beta }^2}{2}\cdot |{\mathrm{\Phi }}_x(s,x){|}^2$$ whose weight is dampened as $s$ increases towards $s=1$.

This picture was helpful to have. It plots the evolution of $\mathrm{\Phi }(s,x)$ from $s=0\to 1$

where, on the right, the red contour corresponds to $s=0$, the navy contour corresponds to $s=1$, and $\mu (t)$ is given by the CDF of a Gaussian centered at $\frac{1}{2}$ with variance $\frac{1}{100}$

When thinking of $\mu (1-s)$, consider the above picture as “backwards” i.e. think of $t$ progressing from $1\to 0$.

Material

• [pdf] Scanned presentation notes

• [demo] [source] Plotting Parisi’s equations

• [link] Scribe notes – Juspreet Singh Sandhu (see Day 2)