$e=e_k=k/(k+1),\ \ k\ge 1$; see [@PR] and references therein, and Figure 1. .3truein

![The microcanonical phase space $R$, outlined in solid lines[]{data-label="phasefig"}](phase_space1h.eps){width="3in"}

We are interested in the relative density of graphs in $R$, more precisely in the entropy, the exponential rate of growth of the number of graphs as $n$ grows, as follows. First consider $$s^{n,\delta}_{e,t}=\frac{\ln(Z^{n,\delta}_{e,t})}{n^2}, \hbox{ and
    then } 
s_{e,t}=\lim_{\delta\to 0^+}\lim_{n\to \infty}s^{n,\delta}_{e,t}.$$ (The existence of the double limit will be proven.) We will measure the growth rate by the entropy density $s_{e,t},$ and the main question of interest for us is the existence of phase transitions ([*i.e.*]{} lack of analyticity of $s_{e,t}$) near the lower boundary of $R$ in Figure 1. The lower boundary consists of the scalloped curve together with the ‘first scallop’, the line from $(0,0)$ to $(1/2,0)$.

We now need to review some notation and results concerning graphons, as recently developed in [@LS1; @LS2; @BCLSV; @BCL; @LS3]. See also the recent book [@Lov].

Graphons
========

Consider the set ${{\mathcal W}}$ of all symmetric, measurable functions $$g:(x,y)\in [0,1]^2\to g(x,y)\in [0,1].$$ Think of each axis as a continuous set of vertices of a graph. For a graph $G\in \hat G^n$ one associates $$\label{checkerboard} g^G(x,y) = \begin{cases} 1 &\hbox{if 