Weakly Self-Avoiding Walk and Supersymmetric RG

Continuous-time weakly self-avoiding walk on $\mathbb Z^4$ has an exact supersymmetric functional-integral representation. The Grassmann sector cancels closed bosonic loops and normalizes the integral; finite-range RG then proves susceptibility exponent $1/4$ and critical two-point decay $|x|^{-2}$ at weak self-repulsion. This auxiliary supersymmetry is a probabilistic device, not spacetime supersymmetry.

Required background. Finite-Range Decompositions and Multiscale Integration, Polymer Activities and Normed RG Coordinates, and Stable Manifolds and Relevant–Marginal Control supply the three analytic inputs of the proof.

Helpful background. Critical φ⁴ Models and Logarithmic Corrections supplies the $n=0$ comparison. Constructive Existence by Model, Dimension, and Observable supplies the model-and-observable status distinction.

Walk and supersymmetric field integral

Let $X(t)$ be continuous-time simple random walk and define its self-intersection local time

$$I(T)=\int_0^T\!\int_0^T \mathbf1_{X(s)=X(t)}\,ds\,dt.$$

For $g>0$, the two-point function and susceptibility are

$$G_{g,\nu}(a,b)=\int_0^\infty \mathbb E_a\!\left[e^{-gI(T)}\mathbf1_{X(T)=b}\right] e^{-\nu T}\,dT, \qquad \chi(g,\nu)=\sum_bG_{g,\nu}(0,b).$$

On a finite set, introduce complex bosons $\phi_x,\bar\phi_x$ and Grassmann one-forms $\psi_x,\bar\psi_x$, and set

$$\tau_x=\phi_x\bar\phi_x+\psi_x\bar\psi_x.$$

With a consistently oriented Berezin measure, the walk two-point function is represented by an integral with action

$$S=\sum_x\left(\tau_{\Delta,x}+g\tau_x^2+\nu\tau_x\right)$$

and insertion $\bar\phi_a\phi_b$. The exact formula, including continuous-time local times and the differential-form convention, is developed in Brydges, Imbrie, and Slade 2009, §§ 4–5, pp. 47–57. Supersymmetric localization makes the partition function without insertions equal to one. This cancellation is what removes vacuum loops; it is not an assumption that the walk has physical fermionic states.

The four-dimensional results

There is a critical killing rate $\nu_c(g)$ for $g>0$ sufficiently small. With $\varepsilon=\nu-\nu_c(g)\downarrow0$,

$$\chi(g,\nu_c+\varepsilon) =A_g\varepsilon^{-1} \bigl(\log\varepsilon^{-1}\bigr)^{1/4} \bigl(1+o(1)\bigr).$$

This is Bauerschmidt, Brydges, and Slade 2015, Theorem 1.1, pp. 822–824. At criticality,

$$G_{g,\nu_c}(0,x) =\frac{c_g}{|x|^2}\bigl(1+o(1)\bigr),$$

with the precise lattice-distance and amplitude convention of Bauerschmidt, Brydges, and Slade 2015, Theorem 1.1, pp. 172–174. The susceptibility logarithm is the $n=0$ value $(n+2)/(n+8)=1/4$ of the scalar flow, while the leading critical two-point function has anomalous exponent $\eta=0$.

The RG proof carries bosonic and fermionic activities in the same normed algebra. Finite-range integration preserves supersymmetry. Localization extracts $g_j,\nu_j,z_j$ and observable source couplings; the polymer remainder remains $O(g_j^3)$. Tuning $\nu_0$ gives the critical orbit, and differentiating observable coordinates reconstructs $G$ and $\chi$.

For the two-point function, the locations $a$ and $b$ introduce a geometric scale that is absent from the vacuum flow. Let $j_{ab}$ be the first scale at which a covariance block can connect the two sites, so $L^{j_{ab}}$ is comparable to $|a-b|$. Below $j_{ab}$ the two linear source couplings renormalize separately. At and above it, a mixed source coordinate $q_{ab,j}$ is generated with a recurrence of the form

$$q_{ab,j+1}-q_{ab,j} =\lambda_{a,j}\lambda_{b,j}C_{j+1}(a,b)+r_{ab,j}.$$

Finite range makes the leading term vanish before coalescence, while differentiated polymer estimates make $\sum_j|r_{ab,j}|$ smaller than the leading Green-function contribution. Summing the covariance pieces then recovers $(-\Delta)^{-1}_{ab}\asymp |a-b|^{-2}$ with a renormalized amplitude. This telescoping observable flow, proved with the bulk domain estimates in Bauerschmidt, Brydges, and Slade 2015, §§ 4–5, pp. 177–188, is why vacuum normalization alone is not enough to establish the critical two-point theorem.

This supplies the concrete method behind Replica and Supersymmetry Methods for Disorder: map the walk exactly to a boson–fermion $|\phi|^4$ integral, tune the killing rate, and derive the proved logarithm. Unlike the replica trick, the finite-volume supersymmetric representation here is an identity before any RG approximation.

Localization check and adversarial imbalance

For a supersymmetric function $F(\tau)$ with sufficient decay, localization reduces its integral to the value at the origin. Applied to the vacuum integrand, this gives normalization one. Differentiating with respect to $\nu$ independently matches integrated local time on the walk side with insertion of $\sum_x\tau_x$ on the field side.

This check should be performed before taking infinite volume. On a finite torus with $\nu>0$, both the time integral on the walk side and the bosonic integral on the field side are absolutely convergent, so differentiation and the algebraic cancellation are justified. Uniform RG bounds are then used to pass to the thermodynamic limit and, separately, to $\nu\downarrow\nu_c$. Reversing that order without a bound can hide an infrared divergence. The finite-volume identity therefore supplies the exact starting object, while the RG theorem supplies the limiting operations; neither part replaces the other.

The sign conventions can be checked in the free case $g=0$. Integrating the quadratic boson–fermion action gives the resolvent $(-\Delta+\nu)^{-1}_{ab}$ for the insertion, whereas the determinant factors cancel in the vacuum normalization. If a proposed Berezin orientation produces the negative resolvent or a determinant prefactor, it cannot represent the nonnegative walk two-point function.

Now replace $\tau_x$ by $\phi_x\bar\phi_x+c\psi_x\bar\psi_x$ with $c\neq1$ while keeping the bosonic action fixed. The fermionic determinant no longer cancels the bosonic determinant, the vacuum integral is not normalized to one, and the localization identity used by the proof fails. One may have another field model, but the same walk representation and RG conclusion cannot be imported.

Status boundary

The theorem concerns a lattice probability model in $d=4$, weak $g$, and named observables. It does not construct a Lorentzian supersymmetric QFT, prove results for strong self-repulsion, or transfer the logarithm to three dimensions. The exact field representation licenses the RG analysis only after all bosonic large-field and fermionic derivative norms are controlled.

Exercise

Show directly that $\chi(g,\nu)=\int_0^\infty\mathbb E_0[e^{-gI(T)}]e^{-\nu T}dT$ whenever summation and integration may be interchanged.

Solution

Sum $G_{g,\nu}(0,b)$ over $b$. For each path and $T$, $\sum_b\mathbf1_{X(T)=b}=1$. Tonelli’s theorem applies because the integrand is nonnegative, giving the displayed formula. This also shows that $-\partial_\nu\log\chi$ probes a duration-weighted observable where the derivative exists.

References

• Bauerschmidt, Roland, David C. Brydges, and Gordon Slade. “Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk.” Communications in Mathematical Physics 338 (2015): 169–193. DOI; Open PDF.
• Bauerschmidt, Roland, David C. Brydges, and Gordon Slade. “Logarithmic Correction for the Susceptibility of the 4-Dimensional Weakly Self-Avoiding Walk: A Renormalisation Group Analysis.” Communications in Mathematical Physics 337 (2015): 817–877. DOI; Open PDF.
• Brydges, David C., John Z. Imbrie, and Gordon Slade. “Functional Integral Representations for Self-Avoiding Walk.” Probability Surveys 6 (2009): 34–61. Open PDF.