\(\def\Real{\mathbb{R}}

\def\Int{\mathbb{Z}}

\def\Comp{\mathbb{C}}

\def\Rat{\mathbb{Q}}

\def\Field{\mathbb{F}}

\def\Fun{\mathbf{Fun}}

\def\e{\mathbf{e}}

\def\f{\mathbf{f}}

\def\bv{\mathbf{v}}

\def\blob{\mathcal{B}}

\)

### The Blob

Consider the following planar “spin model”: the state of the system is a function from \(\Int^2\) into \(\{0,1\}\) (on and off states). We interpret the site \((i,j), i,j\in\Int\) as the *plaque*, i.e. the (closed) square given by the inequalities \(s_{ij}:=i-1/2\leq x\leq i+1/2; j-1/2\leq y\leq j+1/2\).

To any configuration \(\eta\) we associate the corresponding active domain,

\[

A_\eta=\bigcup_{(i,j): \eta(i,j)=1} s_{ij}.

\]

We are interested in the statistical ensembles supported by the finite *contractible* active domains – let’s refer to such domains as ** blobs**.…