Large Orders and Strong-Coupling Limit in Functional Renormalization

TL;DR

This work addresses the large-order behavior and strong-coupling limit of the functional renormalization group in a zero-dimensional setting, establishing Borel-summability for a broad class of microscopic couplings and deriving explicit contour-integral representations for the Borel transform and its inverse. It shows that in the strong-coupling limit the FRG flow converges to a universal fixed point independent of fine microscopic details, yielding closed-form expressions for the limiting disorder correlator and a universal function for ${\cal Z}_{\rm FRG}^{\infty}(w)$. By connecting this zero-dimensional FRG formulation to SUSY/replica field theory, the paper demonstrates consistency with field-theoretic descriptions of disordered elastic systems and aligns with known 1-loop results, while providing a nonperturbative framework for understanding universality and strong-coupling behavior. The findings have implications for depinning physics and molecular biophysics (e.g., DNA/RNA peeling), and offer a solid pathway to study the $\epsilon$-expansion in FRG across dimensions.

Abstract

We study the large-order behavior of the functional renormalization group (FRG). For a model in dimension zero, we establish Borel-summability for a large class of microscopic couplings. Writing the derivatives of FRG as contour integrals, we express the Borel-transform as well as the original series as integrals. Taking the strong-coupling limit in this representation, we show that all short-ranged microscopic disorders flow to the same universal fixed point. Our results are relevant for FRG in disordered elastic systems.

Figures (5)

• Figure 1: The different paths and contour integrals. In blue the one used for Eqs. (\ref{['15']}) and (\ref{['166']}), encircling the cut in Eq. (\ref{['C2B-final']}) (green/dashed). In red the path used for the derivation of Eq. (\ref{['C-asymptotics2']}) which passes through $\phi_{\rm SP}$.
• Figure 2: Plot of $[g_w(\phi)/g_w(\phi_{\rm SP}) ]^n$ for $w=0$, $n=100$, with real part in blue (solid) and imaginary part in red (dashed); $\phi= \phi_{\rm SP}+ i y/\sqrt{n}$, as indicated by the red curve on Fig. \ref{['f:The different paths and contour integrals']}. In green (dot-dashed) $\exp(-\frac{g_w"(\phi)}{ g_w(\phi)}\frac{y^2}{2} )|_{\phi=\phi_{\rm SP}}$, whose integration leads to Eq. (\ref{['C-asymptotics2']}).
• Figure 3: Different solutions for $\tilde{\Delta}(w)$, all rescaled to $\tilde{\Delta}(0) =|\tilde{\Delta}'(0)|=1$. From top to bottom: driven particle (DPM) in Gaussian disorder (blue), Eq. (85) of LeDoussalWiese2008a, Eq. (\ref{['25']}) (red, dashed), Sinai model, Eq. (202) of Wiese2021 (black, dot-dashed), and the 1-loop random-field fixed point, Eq. (88) of Wiese2021 (green, solid).
• Figure 4: ${\cal Z}_{\rm FRG}^{\rm B}(w=0,\lambda)$, evaluated in four different ways: (i) explicit sum from derivatives as given in Eqs. (\ref{['118']}) and (\ref{['118bis']}) (blue solid line). The vertical blue-dashed lines indicate its radius of convergence estimated from Eq. (\ref{['C-asymptotics2']}). (ii) the contour integral (\ref{['166']}) (red dots), (iii) the cut integral (\ref{['C2B-final']}) (green dashed), and (iv) a diagonal Padé resummation of the original series (black crosses). Both integral representations work for $\lambda$ larger than the radius of convergence of the series (but are as expected problematic for negative $\lambda$).
• Figure 5: The function $\tilde{\Delta}_{\rm FRG}"(w,\lambda):= 1-{\cal Z}_{\rm FRG}(w \sqrt{\lambda},\lambda)$ for $\lambda=10$, evaluated via: Padé-Borel (black crosses) on the combinatorial series at order 100. (Some glitches appear, and Padé-Borel breaks down for larger $\lambda$; each Padé is constructed at fixed $w$.) Evaluation of the integral (\ref{['C2final']}) (cyan, solid), indistinguishable form an implementation which keeps the erfc of Eq. (\ref{['159']}) (red, dashed). In blue dashed the asymptotic form (\ref{['24']}).