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On p. 28 of Bertrand Delamotte's Introduction to the Nonperturbative Renormalization Group he writes

$k$ [the renormalization scale] acts as an infrared regulator (for $k \neq 0$) somewhat similar to a box of finite size $\sim k^{−1}$. Thus, for $k > 0$, there is no phase transition and thus no singularity in the free energy $\Gamma_k$. At finite $k$, everything is regular and can be power-expanded safely. We can therefore conclude that

  1. the singularities of $\Gamma$ build up as $k$ is lowered and are thus smoothened by $k$ in $\Gamma_k$.
  2. the precursor of the critical behavior should already show up at finite $k$ for $|q| \gg k$.

Could someone explain the analogy between $k$ as an infrared regulator and a box of finite size $\sim k^{−1}$?

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