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I would like to know if anyone knows about some applications/models where those threshold functions from random graph theory, defined by $$ \lim_{n \to \infty} P(\mathbb{G}_{n,p} \in \mathcal{F}) = \begin{cases} 0 \text{ if } p/p_c \to 0 \\ 1 \text{ if } p/p_c \to \infty \end{cases} $$ are used for. As they describe some kind of phase transition I could imagine they could be used for modelling physical phase transitions (like water to ice etc.) but I haven't encountered a model so far.

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    $\begingroup$ The more physics-related cases are problems of bond percolation, which are a similar model to $\mathbb G_{n,p}$, but with a different host graph the edges are sampled from (usually, a lattice graph). $\endgroup$
    – Misha Lavrov
    Commented Jan 29 at 21:01

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