Infinite-volume nonanalytic-pressure criticality

Let $I\subseteq \mathbb{R}$ be a nonempty open interval of values of a scalar control parameter $\theta$, expressed in one fixed unit if it is dimensional. A function $g:I\to \mathbb{R}$ is real analytic if for every ${\theta }_0\in I$ there are $ϵ>0$ and real coefficients $(a_k{)}_{k\ge 0}$ such that $({\theta }_0-ϵ,{\theta }_0+ϵ)\subseteq I$, the series ${\sum }_{k=0}^{\mathrm{\infty }}{a}_k(\theta -{\theta }_0{)}^k$ converges there, and its sum equals $g(\theta )$ throughout that interval. Let $(N_n{)}_{n\ge 1}$ be positive integers with $N_n\to \mathrm{\infty }$, interpreted as finite system sizes such as site counts. A finite-volume partition-function sequence on $I$ is a sequence of real-analytic functions $Z_n:I\to (0,\mathrm{\infty })$. For each $n$, define its dimensionless log partition density per site by $p_n(\theta ):=N_n^{-1}\log Z_n(\theta )$, where $\log$ is the natural logarithm. If the pointwise limit $p(\theta ):={\lim }_{n\to \mathrm{\infty }}{p}_n(\theta )$ exists as a finite real number for every $\theta \in I$, call $p$ the infinite-volume pressure associated with this sequence. A point ${\theta }_c\in I$ is critical under the nonanalytic-pressure criterion exactly when there do not exist $ϵ>0$ with $({\theta }_c-ϵ,{\theta }_c+ϵ)\subseteq I$ and real coefficients $(a_k{)}_{k\ge 0}$ such that ${\sum }_{k=0}^{\mathrm{\infty }}{a}_k(\theta -{\theta }_c{)}^k$ converges and equals $p(\theta )$ throughout that interval.

This definition specifies only the nonanalytic-pressure criterion and makes no claim that a particular model satisfies it.

The natural logarithm as the inverse of the exponential function