Consider the 2D classical Ising model. It's understood that there is a critical temperature $T_c$, and that the correlation length $\xi(T)$ defined by: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T} := \langle \sigma_i \sigma_j \rangle_T - \langle \sigma_i \rangle_T \langle \sigma_j \rangle_T \to Z^2 \mathrm{exp}\left(- \frac{|i-j|}{\xi(T)}\right)$$ is finite away from $T_c$ but diverges as $1/(T - T_c)$ near the critical temperature. At the critical temperature, it's clear that we should expect scale invariance as the correlation length diverges, formally I expect that: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T_c} \approx \frac{Z^2}{|i - j|^\alpha}$$ for some $\alpha$ (and the $\approx$ is accounting for lattice artifacts at small separations?) What I'm confused about is considering the correlation function: $$f(|i-j|) = \lim_{T \to T_c} \langle \sigma_{\xi(T) \cdot i } \sigma_{\xi(T) \cdot j} \rangle_{\mathrm{connected},T}$$ What is the behaviour of $f(|i-j|)$? Is it exponentially decreasing, or power-law?

(Apologies if this is a duplicate, I couldn't find any times this was asked previously)

Consider the 2D classical Ising model. It's understood that there is a critical temperature $T_c$, and that the correlation length $\xi(T)$ defined by: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T} := \langle \sigma_i \sigma_j \rangle_T - \langle \sigma_i \rangle_T \langle \sigma_j \rangle_T \to Z^2 \mathrm{exp}\left(- \frac{|i-j|}{\xi(T)}\right)$$ is finite away from $T_c$ but diverges as $1/(T - T_c)$ near the critical temperature. At the critical temperature, it's clear that we should expect scale invariance as the correlation length diverges, formally I expect that: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T_c} \approx \frac{Z^2}{|i - j|^\alpha}$$ for some $\alpha$ (and the $\approx$ is accounting for lattice artifacts at small separations?) What I'm confused about is considering the correlation function: $$f\left(i,j\right) = \lim_{T \to T_c} \frac{\langle \sigma_{\xi(T) \cdot i } \sigma_{\xi(T) \cdot j} \rangle_{\mathrm{connected},T}}{\langle \sigma_{\xi(T) \cdot i_0 } \sigma_{\xi(T) \cdot j_0} \rangle_{\mathrm{connected},T}}$$ where $i_0,j_0$ are constant lattice sites (required so that $f$ is nonzero(?)) What is the behaviour of $f(i,j)$, for large $|i-j|$? Is it exponentially decreasing, or power-law?

(Apologies if this is a duplicate, I couldn't find any times this was asked previously)

Consider the 2D classical Ising model. It's understood that there is a critical temperature $T_c$, and that the correlation length $\xi(T)$ defined by: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T} := \langle \sigma_i \sigma_j \rangle_T - \langle \sigma_i \rangle_T \langle \sigma_j \rangle_T \to Z^2 \mathrm{exp}({-\xi(T) |i-j|})$$ is finite away from $T_c$ but diverges as $1/(T - T_c)$ near the critical temperature. At the critical temperature, it's clear that we should expect scale invariance as the correlation length diverges, formally I expect that: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T_c} \approx \frac{Z^2}{|i - j|^\alpha}$$ for some $\alpha$ (and the $\approx$ is accounting for lattice artifacts at small separations?) What I'm confused about is considering the correlation function: $$f(|i-j|) = \lim_{T \to T_c} \langle \sigma_{\xi(T) \cdot i } \sigma_{\xi(T) \cdot j} \rangle_{\mathrm{connected},T}$$ What is the behaviour of $f(|i-j|)$? Is it exponentially decreasing, or power-law?

(Apologies if this is a duplicate, I couldn't find any times this was asked previously)

Consider the 2D classical Ising model. It's understood that there is a critical temperature $T_c$, and that the correlation length $\xi(T)$ defined by: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T} := \langle \sigma_i \sigma_j \rangle_T - \langle \sigma_i \rangle_T \langle \sigma_j \rangle_T \to Z^2 \mathrm{exp}\left(- \frac{|i-j|}{\xi(T)}\right)$$ is finite away from $T_c$ but diverges as $1/(T - T_c)$ near the critical temperature. At the critical temperature, it's clear that we should expect scale invariance as the correlation length diverges, formally I expect that: $$\langle \sigma_i \sigma_j \rangle_\mathrm{connected., T_c} \approx \frac{Z^2}{|i - j|^\alpha}$$ for some $\alpha$ (and the $\approx$ is accounting for lattice artifacts at small separations?) What I'm confused about is considering the correlation function: $$f(|i-j|) = \lim_{T \to T_c} \langle \sigma_{\xi(T) \cdot i } \sigma_{\xi(T) \cdot j} \rangle_{\mathrm{connected},T}$$ What is the behaviour of $f(|i-j|)$? Is it exponentially decreasing, or power-law?

(Apologies if this is a duplicate, I couldn't find any times this was asked previously)