Stable Manifolds and Relevant–Marginal Control

A center-stable manifold for an RG map is a local graph selecting the relevant initial coordinates whose forward orbit remains in a controlled neighborhood while marginal and irrelevant coordinates evolve. Its construction needs a spectral splitting, nonlinear derivative bounds, and a mechanism for the marginal drift; linearized arrows alone neither produce the graph nor describe a global critical surface.

Required background. Renormalized Trajectories and Counterterm Tuning supplies the mixed boundary condition. Polymer Activities and Normed RG Coordinates supplies the irrelevant Banach coordinate.

Helpful background. Clustering, Vacuum Uniqueness, and Mass-Gap Implications explains what a massive limiting trajectory could imply after reconstruction. Counterexamples, Nonconverses, and Hypothesis Stress Tests supplies the local-to-global warning.

Spectral splitting and graph theorem

Let $\mathcal X=E_u\oplus E_c\oplus E_s$ be a Banach space and

$$\mathcal R(x)=A x+N(x), \qquad N(0)=DN(0)=0.$$

Assume $A$ preserves the splitting, $\|A_s\|\leq q<1$, and $A_u$ is invertible with $\|A_u^{-1}\|\leq q$. The center multiplier has modulus one, so a generic hyperbolic stable-manifold theorem is insufficient. For the four-dimensional scalar system the center coordinate instead obeys

$$g_{j+1}=g_j-\beta_jg_j^2+r_j, \qquad \beta_j\geq\beta_*>0,\quad |r_j|\leq Cg_j^3,$$

until the mass scale. Hence $g_j\asymp (g_0^{-1}+\sum_{k<j}\beta_k)^{-1}$ and decays only as $1/j$.

Under weighted Lipschitz and differentiability bounds on the nonlinear terms, the mixed-boundary problem has a unique small solution. Equivalently, there is a local graph

$$W^{cs}_{\mathrm{loc}} =\{x_c+x_s+h(x_c,x_s): (x_c,x_s)\ \text{small}\},$$

with $h:E_c\oplus E_s\to E_u$, such that forward orbits on the graph remain in the prescribed scale-dependent domain. Bauerschmidt, Brydges, and Slade prove the needed nonautonomous, nonhyperbolic stability theorem for the RG form in Bauerschmidt, Brydges, and Slade 2015, Theorem 1.4, pp. 1040–1044. Their proof formulates the orbit as an ODE in a Banach space of weighted sequences and applies an inverse-function argument.

Four-dimensional n-component |φ|⁴

For the weak $n$-component lattice model in $d=4$, use local coordinates $(g,z,\mu)$ plus the polymer remainder $K$. Near the Gaussian fixed point, the mass coordinate has leading multiplier $L^2$ and is relevant; the quartic coordinate is marginally irrelevant for $g>0$; the normalized polymer coordinate contracts. Field-strength and vacuum coordinates are fixed by the chosen normalization and boundary conditions.

The local graph selects critical initial functions $\mu_0^c(g_0)$ and $z_0^c(g_0)$ so that the orbit remains small and $K_j=O(g_j^3)$. To the first controlled orders,

$$g_{j+1}=g_j-\beta_jg_j^2+O(g_j^3), \qquad \mu_0^c(g_0)=O(g_0),\quad z_0^c(g_0)=O(g_0).$$

The coefficients of the critical functions depend on coordinates; their existence and regularity, not those schematic leading orders alone, are the invariant content. This supplies the rigorous construction associated with Fixed Points and Linearized RG Flow: isolate relevant mass and marginal quartic directions, then construct the center-stable graph in the proved neighborhood. The application to the $|\phi|^4$ and weakly self-avoiding-walk flows is explained in Bauerschmidt, Brydges, and Slade 2019, Chapters 6 and 8, pp. 89–104 and 123–138.

Proof mechanism and checks

The Lyapunov–Perron form solves stable components forward and unstable components backward:

$$x_j^s=A_s^jx_0^s+\sum_{k<j}A_s^{j-k-1}P_sN(x_k),$$ $$x_j^u=-\sum_{k\geq j}A_u^{j-k-1}P_uN(x_k).$$

Weighted sup norms make both sums contract. The center equation supplies additional weights through $g_j$. Substituting the fixed point back into the recurrence is one check; differentiating the invariance equation

$$\mathcal R(W^{cs}_{\mathrm{loc}})\subset W^{cs}_{\mathrm{loc}}$$

is another. The tangent of the graph at the fixed point must be $E_c\oplus E_s$.

A quantitative graph check

The spectral inequalities alone do not suffice. On a ball of radius $\rho$, one also needs a bound such as $\|DN(x)\|\leq M\rho$, with $M\rho$ small compared with the gap between contraction in $E_s$ and backward contraction in $E_u$. That condition makes the Lyapunov–Perron operator contract in a weighted sequence norm. In a scale-dependent RG problem, the same comparison is made after inserting weights adapted to $g_j$; a uniform unweighted Lipschitz estimate is usually too crude near a marginal direction.

A two-coordinate model exposes the graph equation. Consider

$$u_{j+1}=2u_j+s_j^2, \qquad s_{j+1}=\tfrac12s_j.$$

Seeking an invariant stable graph $u=h(s)=as^2$ gives

$$h(s/2)=2h(s)+s^2, \qquad \frac a4=2a+1,$$

so $h(s)=-4s^2/7$. Along this graph, both coordinates tend to zero. If the initial point is displaced by $\delta u$ in the unstable direction, then the displacement after $j$ steps is $2^j\delta u$ and eventually exits every fixed local ball. This explicit test illustrates both claims of the theorem: nonlinear terms bend the stable set away from the linear subspace, and the graph selects a unique relevant coordinate locally. It also shows why verifying only $D\mathcal R(0)$ cannot determine the nonlinear critical surface.

Failure and status boundaries

Add an extra relevant operator but omit its coordinate from $E_u$. Its component grows under iteration and the proposed graph is not invariant. Likewise, leave the ball where $DN$ is small: the graph transform or inverse-function estimate loses contraction. Extending the linear arrows beyond that boundary is only a drawing.

The theorem is local. It does not show that every microscopic action enters the neighborhood, that the graph is the complete phase boundary, or that the associated continuum observables exist. Those require global entrance estimates and observable convergence.

Exercise

For $g_{j+1}=g_j-\beta g_j^2$ with $g_j>0$, show heuristically why $g_j\sim(\beta j)^{-1}$.

Solution

Compute $g_{j+1}^{-1}-g_j^{-1}=\beta/(1-\beta g_j)=\beta+O(g_j)$. Summing gives $g_j^{-1}=g_0^{-1}+\beta j+O(\log j)$ while the orbit remains small. Hence $g_j\sim1/(\beta j)$. A rigorous proof controls the accumulated remainder and scale dependence of $\beta_j$.

References

• Bauerschmidt, Roland, David C. Brydges, and Gordon Slade. Introduction to a Renormalisation Group Method. Lecture Notes in Mathematics 2242. Singapore: Springer, 2019. DOI; Open PDF.
• Bauerschmidt, Roland, David C. Brydges, and Gordon Slade. “Structural Stability of a Dynamical System Near a Non-Hyperbolic Fixed Point.” Annales Henri Poincaré 16 (2015): 1033–1065. Open PDF.