Renormalization of unbounded stretched graphons

Proof. Consider any finite graph \(G\) with the vertex set \(V(G)\), such that \(|V(G)|=n\) and each vertex \(v_{i}\) is weighted by some \(\alpha_{i} \in \mathbb{R}\), and the edge set \(E(G)\) such that any edge \(v_{i}v_{j}\) is weighted by \(\beta_{ij} \in \mathbb{R}\). For any partition \(\sigma:=(I_{1},…,I_{n})\) of \([0,1]\) with \({m}(I_{k})=\frac{|\alpha_{k}|}{\sum\limits_{i=1}^{n}|\alpha_{i}|}\), the graph function \[W^{\sigma}_{G}(x,y)=\bigg\{_{0, \ {otherwise}}^{\beta_{ij}-\lfloor \beta_{ij} \rfloor \ , \ (x,y)\in I_{i} \times I_{j}}, \tag{4}\] is a labeled graphon associated to \(G\). While relabeling for the graph is given by permutation of its vertices, any measure preserving bijection \(\rho\) on \([0,1]\) defines a relabeling \(W^{\rho(\sigma)}\) associated to \(G\) such that \(\rho(\sigma):=(\rho(I_{1}),…,\rho(I_{n}))\) is a partition of \([0,1]\). Up to the weakly isomorphism, the unique class \[W_{G}:=[W^{\sigma}_{G}]_{\approx}=\bigg\{ W^{\rho(\sigma)} \ : \ \rho \ \ \text{Lebesgue measure presering transformation}\bigg\}, \tag{5}\] is called the canonical unlabeled graphon associated to \(G\). ◻

Proof. On the one hand, any Feynman diagram \(\Gamma\) has a rooted tree (or forest) representation \(t_{\Gamma}\) such that the loop number of \(\Gamma\) identifies the number of vertices of \(t_{\Gamma}\). Each vertex \(v_{\gamma}\) in \(t_{\Gamma}\) is decorated by a (1PI) primitive Feynman subdiagram \(\gamma\) of \(\Gamma\). For 1PI primitive Feynman subdiagrams \(\gamma_{1},\gamma_{2}\) of \(\Gamma\), if \(\gamma_{1} \subseteq \gamma_{2}\), then there exists an edge from the vertex \(v_{\gamma_{1}}\) to the vertex \(v_{\gamma_{2}}\) in \(t_{\Gamma}\). If \(\gamma_{1} \not \subseteq \gamma_{2}\) and \(\gamma_{2} \not \subseteq \gamma_{1}\), then there is no edge between vertices \(v_{\gamma_{1}}\) and \(v_{\gamma_{2}}\). If \(\Gamma\) has overlapping loops, then \(t_{\Gamma}\) is a linear combination of decorated non-planar tooted trees. Therefore the renormalization Hopf algebra \(H_{FG}(\Phi)\) embeds, by an injective Hopf algebra homomorphism, in a decorated version of the Connes–Kreimer Hopf algebra presented by \(H_{CK}(\Phi)\) such that vertices of rooted trees are decorated by (1PI) primitive Feynman (sub)diagrams.

On the other hand, let \(B^{+}\) be a linear operator on \(H_{CK}\) which maps each forest \(t_{1}…t_{n}\) to a new non-planar rooted tree \(t\) by adding a new root \(\bullet\) together with \(n\) new edges which connect \(\bullet\) to the roots of \(t_{1}\),…, \(t_{n}\). For any \(t=B^{+}(t_{1}…t_{n})\), we get \(\Delta_{ren}(t) = ({id} \otimes B^{+}) \Delta_{ren}(t_{1}…t_{n}) + t \otimes \mathbb{I}\). Thanks to the Hochschild cohomology of commutative Hopf algebras, consider a category of pairs \((H,L)\) of a commutative (graded) Hopf algebra and a linear operator \(L:H \rightarrow H\) which satisfies the equation \[\Delta_{H}L(x) = ({id} \otimes L) \Delta_{H}(x) + L(x) \otimes \mathbb{I} \ . \tag{13}\]

Morphisms of this category are Hopf algebra homomorphisms \(f:H_{1} \rightarrow H_{2}\) such that \(L_{2} \circ f = f \circ L_{1}\). The object \((H_{ CK},B^{+})\) is the universal object. It means that for any object \((H,L)\), there exists a unique Hopf algebra homomorphism \(f_{H}:H_{CK} \rightarrow H\) such that \(L \circ f_{H} = f_{H} \circ B^{+}\).

The injective Hopf algebra homomorphism \(\Gamma \mapsto t_{\Gamma}\) together with the unique Hopf algebra homomorphism \(f_{H_{{FG}}(\Phi)}: H_{CK} \rightarrow H_{{FG}}(\Phi)\) determine the quotient Hopf algebra \(H_{CK}(\Phi)/I_{\Phi}\) for some Hopf ideal \(I_{\Phi}\) as the universal representation of \(H_{{FG}}(\Phi)\). ◻

Proof. Thanks to Lemma 1, the adjacency matrix of any \(t \in H_{CK}\) determines its unique canonical unlabeled graphon \(W_{t} \in \mathcal{W}_{\approx}^{([0,1),{m})}(\mathbb{R})\). For a fixed \(p \ge 1\), we apply (7) to define a distance between non-planar rooted trees given by \[\label{metr-1} d_{p, cut}(s,t)=d_{cut}\left(\frac{W_{s}}{||W_{s}||_{p,cut}},\frac{W_{t}}{||W_{t}||_{p,cut}}\right) . \tag{14}\]

The graph limits of all Cauchy sequences in \(H_{CK}\) are represented by some elements in \(\mathcal{W}_{\approx}^{([0,1),{m})}(\mathbb{R})\). Consider any recursive equation \(X = \mathbb{I} + \sum\limits_{n=1}^{\infty} c^{n} B^{+}_{\bullet}(X^{n+1})\) with \(c > 1\) in \(H_{CK}\) such that \(X=\sum\limits_{n=0}^{\infty}c^{n}X_{n}\) is given by the recursive relations \[X_{n} = \sum\limits_{j=1}^{n} B^{+}_{\bullet} \bigg(\sum\limits_{k_{1}+…+k_{j+1}=n-j,k_{i}\ge 0} X_{k_{1}}…X_{k_{j+1}} \bigg) \ , \ X_{0}=\mathbb{I} \ . \tag{15}\]

It can be presented by a stretched graphon \(W_{X} \in \mathcal{W}_{\approx}^{([0,1),{m})}(\mathbb{R})\) such that \[{lim}_{m \rightarrow \infty} Y_{m} = X \Leftrightarrow {lim}_{m \rightarrow \infty} \frac{W_{Y_{m}}}{||W_{Y_{m}}||_{p,cut}} = W_{X}, \tag{16}\] where \(Y_{m}=\sum\limits_{k=1}^{m}c^{k}X_{k}\). The stretched graphon \(W_{Y_{m}} \in \mathcal{W}_{\approx}^{([0,1),{m})}(\mathbb{R})\) is determined by stretched graphons \(W_{X_{j}}\) defined on subintervals \(I_{j} \subset [0,1)\) with \({m}(I_{j})=\frac{c^{j}}{\sum\limits_{k=1}^{m}c^{k}}\).

For any \(x \in H_{CK} \otimes H_{CK}\), \[\label{cross-100} ||x||_{p, cross} = {inf} \ \bigg\{\sum\limits_{i=1}^{n} ||W_{s'_{i}}||_{p, cut} ||W_{s''_{i}}||_{p, cut} \ , \ x=\sum\limits_{i=1}^{n} s'_{i} \otimes s''_{i} \bigg\}, \tag{17}\] defines a norm structure such that graph limits of Cauchy sequences in \(H_{CK} \otimes H_{CK}\) are represented by some elements in \(\mathcal{W}_{\approx}^{([0,1),{m})}(\mathbb{R})\). It leads us to describe \(\Delta_{ren}\) as a linear bounded operator between completed normed spaces which should be continuous. Therefore for any Cauchy sequence \(\{t_{n}\}_{n \ge 1}\) in \(H_{CK}\) which converges to the graph limit \(X\), \(\Delta_{ren}(X)\) and \(S_{ren}(X)\) are defined as graph limits of the sequences \(\{\Delta_{ren}(t_{n})\}_{n \ge 1}\) and \(\{S_{ren}(t_{n})\}_{n \ge 1}\) such that \[{lim}_{n \rightarrow \infty} \frac{W_{\Delta_{ren}(t_{n})}}{||W_{\Delta_{ren}(t_{n})}||_{p,cut}} = W_{\Delta_{ren}(X)} \ , \ {lim}_{n \rightarrow \infty} \frac{W_{S_{ren}(t_{n})}}{||W_{S_{ren}(t_{n})}||_{p,cut}} = W_{S_{ren}(X)} \ . \tag{18}\] ◻

Proof. The zero graphon \(W_{\mathbb{I}}\) is its unit. For any stretched graphon \(W\) defined on \(I \times I \subseteq [0,\infty) \times [0,\infty)\), consider the equation \(W + W' = W_{\mathbb{I}}\) such that \(W+W': I \bigsqcup I' \times I \bigsqcup I' \rightarrow \mathbb{R}\) with \(I \cap I' = \emptyset \ , \ { m}(I)={m}(I')=1\). We apply Lebesgue measure preserving transformations \(x \mapsto x-\alpha\) and \(x \mapsto nx \ ({mod \ 1})\) to project \(I'\) onto \(I\) or vice versa to replace the direct sum with its corresponding pointwise summation. Therefore \(W+W'\), which is weakly isomorphic to \(W_{\mathbb{I}}\), is also weakly isomorphic to the pointwise summation \(W(x,y)+W'(x,y)\) on \(I \times I\) or on \(I' \times I'\). It means that \(W(x,y)+W'(x,y)=0\) (i.e. \(W'=-W\)) Lebesgue almost everywhere on \(I \times I\) or on \(I' \times I'\). ◻

Proof. For direct sums \(U,V\) defined on \(\bigsqcup_{n=1}^{\infty} I_{n} \times \bigsqcup_{n=1}^{\infty} I_{n}\) and \(\bigsqcup_{n=1}^{\infty} J_{n} \times \bigsqcup_{n=1}^{\infty} J_{n}\), and also \(K:=\bigsqcup_{n=1}^{\infty} I_{n} \bigcap \bigsqcup_{n=1}^{\infty} J_{n}\), we have the following situations.

• If \((x,y) \in K \times K\), then \((U-V)(x,y)=U(x,y) – V(x,y)\). (i.e. pointwise summation)

• If \((x,y) \in \bigsqcup_{n=1}^{\infty} I_{n} \setminus K \times \bigsqcup_{n=1}^{\infty} I_{n} \setminus K\), then \((U-V)(x,y)=U(x,y)\),

• If \((x,y) \in \bigsqcup_{n=1}^{\infty} J_{n} \setminus K \times \bigsqcup_{n=1}^{\infty} J_{n} \setminus K\), then \((U-V)(x,y)=V(x,y)\).
  

Therefore \[\begin{aligned} d_{p,cut}(U,V)=&\bigg|\bigg|U+(-V)\bigg|\bigg|_{p,cut}\notag\\ =& \bigg|\bigg|U|_{\bigsqcup_{n=1}^{\infty} I_{n} \setminus K \times \bigsqcup_{n=1}^{\infty} I_{n} \setminus K}\bigg|\bigg|_{p,cut} + \bigg|\bigg|-V|_{\bigsqcup_{n=1}^{\infty} J_{n} \setminus K \times \bigsqcup_{n=1}^{\infty} J_{n} \setminus K}\bigg|\bigg|_{p,cut} + \bigg|\bigg|(U-V)|_{K \times K}\bigg|\bigg|_{p,cut}. \end{aligned} \tag{20}\] ◻

Proof. Consider the sequence \(\{U^{(m)}\}_{m \ge 1}\) of its corresponding partial direct sums. Thanks to [9], we apply Lebesgue measure preserving transformations on \([0,\infty)\) to project \(U^{(m)}\) to its corresponding stretched graphon \(\tilde{U}^{(m)}: \bigsqcup_{k=1}^{m} J_{k} \times \bigsqcup_{k=1}^{m} J_{k} \subseteq [0,1) \times [0,1) \rightarrow \mathbb{R}\) with respect to the stretched graphons \(\tilde{U}_{k}: J_{k} \times J_{k} \rightarrow \mathbb{R}\) for each \(1 \le k \le m\) such that (i) for \(\sum\limits_{k=1}^{m} |c_{k}|<1\), \({m}(J_{k})=\frac{|c_{k}|}{\sum\limits_{k=1}^{m} |c_{k}|}\), (ii) for \(\sum\limits_{k=1}^{m} |c_{k}|>1\), \({m}(J_{k})=\frac{|1/c_{k}|}{\sum\limits_{k=1}^{m} (|1/c_{k}|)}\), and (iii) for \(\sum\limits_{k=1}^{m} |c_{k}|=1\), \({ m}(J_{k})=\frac{2^{-{ val}(U_{k})}}{\sum\limits_{k=1}^{m} 2^{-{val}(U_{k})}}\) with \({val}(U_{k}):= {Max} \ \{n \ge 1: \ U_{k} \in \bigoplus_{l \ge n} \mathcal{H}^{\mathfrak{U},(l)}\}\).

For each \(m \ge 1\), define a random graph \(R_{m}\) with the vertex set \(A_{m}:=\{x_{1},…,x_{m(m+1)/2}\}\) uniformly chosen from nodes in \(\bigsqcup_{n=1}^{m}J_{n}\) such that with the probability \(||\tilde{U}^{(m)}||^{-1}_{1}\tilde{U}^{(m)}(x_{i},x_{j})\) there exists an edge between \(x_{i}\) and \(x_{j}\) in \(R_{m}\). When \(m\) tends to infinity, the sequence \(\{R_{m}\}_{m \ge 1}\) converges in distribution to the stretched graphon \(\tilde{U}:\bigsqcup_{k=1}^{\infty} J_{k} \times \bigsqcup_{k=1}^{\infty} J_{k} \subseteq [0,1) \times [0,1) \rightarrow \mathbb{R}\) which is weakly isomorphic to \(U\). ◻

Proof. Consider the sequence \(\{U^{(m)}\}_{m \ge 1}\) of partial direct sums of \(U\). The probability distribution of any finite graph \(G\) in \(U\) is given in terms of the probability distribution of \(G\) in the partial direct sums \(U^{(m)}\) given by integrating over all possible choices of \(x_{1},…,x_{|V(G)|} \in A_{m}\) with respect to \(U^{(m)}\) such that \(A_{m}\) is determined in the proof of Corollary 1. In other words, \[\label{f-6} \mathbb{P}_{U^{(m)}}(G) = \int_{(\bigsqcup_{k=1}^{m} I_{k})^{|V(G)|}} \prod_{e_{ij} \in E(G)} \frac{U^{(m)}(x_{i},x_{j})}{||U^{(m)}||_{cut}} \prod_{e_{ij} \notin E(G)} \bigg( 1 – \frac{U^{(m)}(x_{i},x_{j})}{||U^{(m)}||_{cut}} \bigg) \ dx_{1}…dx_{|V(G)|} \tag{36}\] \[\Rightarrow \ \mathbb{P}_{U}(G) = {lim}_{m \rightarrow \infty} \mathbb{P}_{U^{(m)}}(G). \tag{37}\] ◻

Proof. The proof is similar to the proof of Theorem 1 for all \(L^{p}\)-stretched graphons which represent some graph limits. The resulting topological Hopf algebra, presented by \(\mathcal{H}^{\mathfrak{U},{cut}}_{p}\), is called \(L^{p}\)-topological core Hopf algebra of stretched graphons. ◻

Proof. The core coproduct is definable for any stretched graphon in \(\mathcal{W}_{\approx}^{\mathfrak{U}}(\mathbb{R})\) with respect to the cut-distance metric (2), but the core coproduct of a \(L^{p}\)-stretched graphon which does not represent a graph limit might fail to be defined. Therefore for any \(p \ge 1\), \(\mathcal{H}^{\mathfrak{U},{cut}}_{p}\) embed in \(\mathcal{H}^{\mathfrak{U},{cut}}\). The rest of the cases are the result of Lemma 3, Theorem 1 and Corollary 3. ◻

Proof. We apply Theorem 1. Let \(W_{X} \in \mathcal{W}^{\mathfrak{U}}_{\approx}(\mathbb{R})\) be any unbounded stretched graphon representation of a graph limit \(X\) with \(||W_{X}||_{cut}=\infty\).

Step I. Regularized character and its meromorphic extension.

The existence of meromorphic continuation to the complex plane for generalized functions with respect to a parameter was shown in [29] and applied for dimensional regularization parameter in [17, 30]. Here we update this technique for our setting. Consider any \(n\) variable polynomial \(Q(u)\) with \(u=(u_{1},…,u_{n})\), and a parameter \(f(z,n) \in \mathbb{C}\) which depends on \(z,n\). Define the module of elements \(p(u,f(z,n))Q(u)^{-f(z,n)/2 – k}\) over the ring of differential operators \(P(u,\partial,f(z,n))\) where \(p(u,f(z,n))\) are polynomials in \(u\) with rational functions of \(f(z,n)\) as their coefficients. Since there exist some \(P\) and \(k\) which contribute to the equation \(Q(u)^{-f(z,n)/2 – k-1} = P(u,\partial,f(z,n)) Q(u)^{-f(z,n)/2 – k}\), this module is finitely generated in terms of the filtrations \[\mathcal{M}_{m}:= \bigg\{p(u,f(z,n))Q(u)^{-f(z,n)/2 – m} \ : \ {deg}(p(u,f(z,n))) \le m({deg}(Q(u))+1)\bigg\}, \tag{41}\] such that \({dim}(\mathcal{M}_{m}) \sim ({deg}(Q(u))+1)^{n}m^{n}\). Therefore we can find polynomials \(T\) and \(T_{j}\) which contribute to the equations \(TQ(u) + \sum\limits_{j} T_{j} \partial_{j} Q(u) = 1\). This leads us to define a new polynomial differential operator \(L(f(z,n))\) in \(n\) variables given by \[\label{meromorphic-1} L(f(z,n)) = -\frac{f(z,n)}{2}T + \sum\limits_{j} T_{j}\partial_{j}, \tag{42}\] and a new polynomial \(q(f(z,n))=-\frac{f(z,n)}{2}\) which satisfy in the equation \[\label{meromorphic-2} L(f(z,n))Q(u)^{-f(z,n)/2} = q(f(z,n)) Q(u)^{-f(z,n)/2 – 1} \ . \tag{43}\]

Consider certain regularized characters \(\phi^{z}\) in \(\mathbb{G}^{\mathfrak{U}}(A_{dr})\) with respect to the parameter \(z \in \partial {\Delta}^{*}\) which varies in the boundary of an infinitesimal punctured disk \({\Delta}^{*}\) around zero in the complex plane. They are homomorphisms which assign a Laurent series to any generator \(W_{X}\) of \(\mathcal{H}^{\mathfrak{U},cut}\). This Laurent series is the result of defining a new function \(\phi^{z}(W_{X})\), which has a meromorphic continuation to the complex plane, such as the one defined in terms of replacing the integrator \(dxdy\) in (2) by \(d^{1-z}xd^{1-z}y\). Thanks to Lemmas 1.7 and 1.8 in [17], these new functions could have general forms such as \[\label{reg-char-1} \phi^{z}(W_{X})= \int_{u=(u_{1},…,u_{n}) \in [0,\infty)^{n}} g(z,u)Q_{W_{X}}(u)^{f(z,n)} \prod_{j=1}^{n}du_{j} \ , \ f(z,n) \in \mathbb{C} \ , \tag{44}\] such that \(Q_{W_{X}}(u)\) is a polynomial in \(n\) variables built in terms of the Schwinger parametrization \[\int \frac{dxdy}{W_{X}(x,y)^{n}} = \frac{1}{\Gamma(n)} \int dxdy \int_{[0,\infty)} du u^{n-1} e^{-uW_{X}(x,y)} \ , \tag{45}\] and \(g(z,u)\) has the Taylor expansion \(\sum\limits_{m \in \mathbb{Z}}g_{m}(u)z^{m}\) at \(z=0\). Firstly, by applying the integration-by-parts together with (42) and (43), we get \[\begin{aligned} \int_{[0,\infty)^{n}} g(z,u)Q_{W_{X}}(u)^{f(z,n)+2} \prod_{j=1}^{n}du_{j} = q(f(z,n))^{-1} \int_{[0,\infty)^{n}} L(f(z,n))^{\ast}g(z,u)Q_{W_{X}}(u)^{f(z,n)} \prod_{j=1}^{n}du_{j} + B(f(z,n)), \end{aligned} \tag{46}\] such that \(L(f(z,n))^{\ast}\) is the adjoint operator of \(L(f(z,n))\) and \(B(f(z,n))\) is a meromorphic boundary term. Secondly, the coefficients \(g_{m}(u)\) of the Taylor expansion of \(g(z,u)\) are of the form \(Q_{W_{X}}(u)^{-|m|}h_{m}\) for some smooth functions \(h_{m}\) of rapid decay defined for \(u_{j} \in [0,\infty)\) whose derivations are also of rapid decay. Therefore, \(\phi^{z}(W_{X})\) can be extended meromorphically to the whole complex plane.

Step II. Renormalization program.

Consider the infinite dimensional complex Lie group \(\mathbb{G}^{\mathfrak{U}}(\mathbb{C}) = {Hom}(\mathcal{H}^{\mathfrak{U},{cut}},\mathbb{C})\) of characters of the topological core Hopf algebra of stretched graphons. It is the projective limit of finite dimensional Lie groups modeled by suitable Lie subgroups of \({GL}_{n}(\mathbb{C})\). Therefore the topology of \(\mathbb{G}^{\mathfrak{U}}(\mathbb{C})\), which is inherited from the inverse system of simpler Lie groups, is the topology of pointwise convergence. This topology is valid for the space of algebra homomorphisms from \(\mathcal{H}^{\mathfrak{U},{ cut}}\) to any commutative locally convex algebra such as \(\mathbb{C}[z^{-1}]\) and \(z^{-1}\mathbb{C}[z^{-1}]\). Since \(\mathcal{H}^{\mathfrak{U},{cut}}\) is connected and graded, \(\mathbb{G}^{\mathfrak{U}}(\mathbb{C})\) becomes a Baker–Campbell–Hausdorff (BCH) Lie group which means that smooth solutions of differential equations are well-defined.

For any complex value \(z\) in the boundary of an infinitesimal punctured disk around \(z=0\) in the complex plane, we work on regularized characters \(\phi^{z} \in \mathbb{G}^{\mathfrak{U}}(A_{dr})\) with the general form (44) such that for any stretched graphon \(W_{X}\), the integral \(\phi^{z}(W_{X})\) is meromorphic in \(z\) where (i) the coefficients of the Laurent series, which grow at most polynomially, are uniformly bounded, or (ii) the location and order of poles in the Laurent series remain stable under cut-distance convergence of stretched graphons. These characters, as linear and bounded maps, are continuous. These continuous characters, which allow for controlled subtraction of divergences, form a dense subspace in \(\mathbb{G}^{\mathfrak{U}}(A_{ dr})\) with respect to the pointwise convergence topology.

Let \(T_{\phi^{z}(W_{X})}\) be the polar part of the Laurent series expansion \(L_{\phi^{z}(W_{X})}\) of \(\phi^{z}(W_{X})\) such that the minimal subtraction map \(R_{ms}\) removes the polar part \(L_{\phi^{z}(W_{X})} – T_{\phi^{z}(W_{X})}\). Following the Atkinson Theorem [26, 27], for the algebra \(A_{dr}\) of Laurent series with finite pole parts, the Rota–Baxter algebra \((A_{dr},R_{ms})\) inherits the (Hopf–)Birkhoff factorization for elements of \(\mathbb{G}^{\mathfrak{U}}(A_{dr})\). The unique Hopf–Birkhoff factorization of any \(\phi^{z} \in \mathbb{G}^{\mathfrak{U}}(A_{dr})\), such as (44), is given by the pair \((\phi^{z}_{-},\phi^{z}_{+})\) of characters on \(\mathcal{H}^{\mathfrak{U},{cut}}\) such that \(\phi^{z} = (\phi_{-}^{z})^{\ast_{core} -1} \ast_{core} \phi^{z}_{+}\) with \(\phi^{z}_{-} \in \mathbb{G}^{\mathfrak{U}}(\mathbb{C}[z^{-1}])\) and \(\phi^{z}_{+} \in \mathbb{G}^{\mathfrak{U}}(z^{-1}\mathbb{C}[z^{-1}])\).

Thanks to [17, 19, 22, 26, 27] and the coproduct (24), we show the compatibility of this factorization with unbounded stretched graphons and infinite direct sums of stretched graphons. For any sequence \(\{W_{n}\}_{n \ge 1}\) of unbounded stretched graphons in \(\mathcal{W}^{\mathfrak{U}}_{\approx}(\mathbb{R})\), there exists a subsequence \(\{U_{n_{i}}\}_{i \ge 1}\) which converges to an unbounded stretched graphon \(V\) with respect to the cut-distance metric (2). We get \(\phi^{z}(V)\) as the limit of the sequence \(\{\phi^{z}(U_{n_{i}})\}_{i \ge 1}\) with respect to the topology of pointwise convergence when \(n\) tends to infinity. The renormalized value for \(V\) is determined by the limits of the sequences \(\{\phi_{-}^{z}(U_{n_{i}})\}_{i \ge 1}\) and \(\{\phi^{z}_{+}(U_{n_{i}})\}_{i \ge 1}\) such that \[\label{f-15} \phi_{+}^{z}(V)=\bigg(\phi_{-}^{z}*_{core}\phi^{z}\bigg)(V) = {lim}_{i \rightarrow \infty} \bigg(\phi_{-}^{z}*_{core}\phi^{z}\bigg)(U_{n_{i}}) \ , \tag{47}\] \[\label{f-151} \phi^{z}_{-}(U_{n_{i}}) = – R_{ms} \bigg(\phi^{z}(U_{n_{i}}) + \sum \phi^{z}_{-}(Z'_{(n_{i})}) \phi^{z}(Z''_{(n_{i})}) \bigg) \ , \tag{48}\] \[\label{f-152} \phi^{z}_{+}(U_{n_{i}})=\phi^{z}(U_{n_{i}}) + \phi^{z}_{-}(U_{n_{i}}) + \sum \phi^{z}_{-}(Z'_{(n_{i})})\phi^{z}(Z''_{(n_{i})}), \tag{49}\] with respect to the Sweedler notation for coproducts. Following a similar process, for any infinite direct sum \(U\) of stretched graphons \(U_{n}\), \(n \ge 1\), with the corresponding sequence \(\{U^{(m)}\}_{m \ge 1}\) of partial direct sums, we get \(\phi^{z}(U)\) as the limit of the sequence \(\{\phi^{z}(U^{(m)})\}_{m \ge 1}\) such that (i) \(\phi^{z}(U)\) has a meromorphic extension to the whole complex plane and (ii) the renormalized value \(\phi^{z}_{+}(U)\) exists as the limit of the sequence \(\{\phi_{+}^{z}(U^{(m)}) \}_{m \ge 1}\). ◻

Proof. It is a result of Corollaries 3, 4 such that we need to update the proof of Theorem 2 for the complex Lie group \(\mathbb{G}^{\mathfrak{U}}_{p}(A_{dr})={Hom}(\mathcal{H}_{p}^{\mathfrak{U},cut},A_{dr})\) of regularized characters on \(\mathcal{H}_{p}^{\mathfrak{U}, cut}\). ◻

Proof. This is a consequence of [32], Theorems 1, 2 and Definition 3. For any unbounded stretched graphon \(U\), as the infinite direct sum of stretched graphons \(U_{n}\) with the corresponding sequence \(\{U^{(m)}\}_{m \ge 1}\) of finite partial direct sums, \(\bigg(\phi \star_{\lambda,\hbar} \psi\bigg)(U)\) is the limit of the sequence \(\bigg\{\bigg(\phi \star_{\lambda,\hbar} \psi\bigg)(U^{(m)}) \bigg\}_{m \ge 1}\) when \(m\) tends to infinity. Then, thanks to Theorem 2, for the regularized character \(\phi^{z}:\mathcal{H}^{\mathfrak{U},{cut}} \rightarrow A_{{dr}}\) given by (44), we get \[\phi_{+}^{z}(U)=\bigg(\phi_{-}^{z} \star_{\lambda,\hbar} \phi^{z} \bigg) (U) =\bigg(\phi_{-}^{z} \star_{\lambda,\hbar} \phi^{z} \bigg)({lim}_{m \rightarrow \infty} U^{(m)}) = {lim}_{m \rightarrow \infty} \bigg(\phi_{-}^{z} \star_{\lambda,\hbar} \phi^{z} \bigg) (U^{(m)}). \tag{52}\] ◻

Proof. This is a consequence of Theorems 1, 2, Definition 3, Lemma 6 and Dubois-Violette’s noncommutative differential calculus [33].

Consider \({Der}_{\lambda}\) as the space of derivations \(\theta:C_{\lambda} \rightarrow C_{\lambda}\) which are linear maps that satisfy Leibniz rule, and \({Ham}^{\lambda}\) as the \(Z(C_{\lambda})\)-module generated by Hamiltonian derivations \({ham}(\phi): \psi \mapsto [\phi,\psi]_{\lambda}\) for any \(\phi \in C_{\lambda}\) such that \(Z(C_{\lambda})\) is the center of \(C_{\lambda}\). For \(n \ge 1\), define \(\Omega^{n}(C_{\lambda})\) as the space of all \(Z(C_{\lambda})\)-multilinear antisymmetric functionals \({Ham}^{\lambda} \times …^{n} … \times {Ham}^{\lambda} \rightarrow C_{\lambda}\) such that for \(n=0\), \(\Omega^{0}(C_{\lambda}) = C_{\lambda}\).

If \(\Omega^{\bullet}(C_{\lambda})=\bigoplus_{n=0}^{\infty} \Omega^{n}(C_{\lambda})\), then for any \(\omega \in \Omega^{\bullet}(C_{\lambda})\) and \(\theta_{0},…,\theta_{n} \in {Der}_{\lambda}\), define \[(d_{\lambda} \omega)(\theta_{0},…,\theta_{n}):= \sum\limits_{k=0}^{n} (-1)^{k} \theta_{k} \omega(\theta_{0},…,\hat{\theta_{k}},…,\theta_{n}) + \sum\limits_{0 \le r < s \le n} (-1)^{r+s} \omega([\theta_{r},\theta_{s}]_{\lambda},\theta_{0},…,\hat{\theta_{r}},…,\hat{\theta_{s}},…,\theta_{n}). \tag{54}\]

The noncommutative deRham complex associated to the differential graded algebra \((\Omega^{\bullet}(C_{\lambda}),d_{\lambda})\) is given by \({ DR}^{\bullet}(C_{\lambda}):= \frac{\Omega^{\bullet}(C_{\lambda})}{[\Omega^{\bullet}(C_{\lambda}),\Omega^{\bullet}(C_{\lambda})]_{\lambda}}\). The symplectic form \[\omega_{\lambda}: {Ham}^{\lambda} \times {Ham}^{\lambda} \rightarrow C_{\lambda}, \ (\theta,\theta') = \bigg(\sum\limits_{i} u_{i} \circ_{\lambda} {ham}(\phi_{i}),\sum\limits_{j} v_{j} \circ_{\lambda} {ham}(\psi_{j})\bigg) \mapsto \sum\limits_{i,j} u_{i} \circ_{\lambda} v_{j} \circ_{\lambda} [\phi_{i},\psi_{j}]_{\lambda}, \tag{55}\] in \(\Omega^{2}(C_{\lambda})\) defines the Poisson structure \(\{\phi,\psi\}_{\lambda} = i_{\theta_{\phi}^{\lambda}} i_{\theta_{\psi}^{\lambda}} \omega_{\lambda}\) on \(C_{\lambda}\) with the corresponding star products \[\phi \star_{\{.,.\}_{\lambda},z} \psi = \phi *_{core} \psi + |z| \{\phi,\psi\}_{\lambda} + O(|z|^{2}), \tag{56}\] such that \(z\) varies in \(\partial {\Delta}^{*}\) and \(\theta_{\phi}^{\lambda},\theta_{\psi}^{\lambda}\) are symplectic vector fields associated with \(d_{\lambda}\phi,d_{\lambda}\psi\).

We have \[\phi_{+}^{z}(U)=\bigg(\phi_{-}^{z} \star_{\{.,.\}_{\lambda},z} \phi^{z} \bigg) (U) =\bigg(\phi_{-}^{z} \star_{\{.,.\}_{\lambda},z} \phi^{z} \bigg)({ lim}_{m \rightarrow \infty} U^{(m)}) = {lim}_{m \rightarrow \infty} \bigg(\phi_{-}^{z} \star_{\{.,.\}_{\lambda},z} \phi^{z} \bigg) (U^{(m)}). \tag{57}\]

Therefore the continuum limit of \(\bigg\{\phi^{z}_{-} *_{core} \phi^{z} + |z|\{\phi^{z}_{-},\phi^{z}\}_{\lambda} + O(|z|^{2}) \bigg\}_{z \rightarrow 0}\) describes the analytic continuation for \(\phi^{z}_{+}(U)\) in \({\Delta}^{\ast}\). ◻

Proof. For each \(n\), define \(U^{(n)}:=U|_{I_{n} \times I_{n}} = W_{1}+…+W_{n}\) with \(I_{n}=\bigg[\sum\limits_{k=1}^{n-1}\frac{1}{2^{k}},\sum\limits_{k=1}^{n}\frac{1}{2^{k}}\bigg)\) such that the total measure of all intervals is less than \(1\) while they converge to \(1\). When \(n\) increases, the mass of \(W_{n}\) concentrates in smaller regions which allows to remain small with respect to the cut-distance topology. It can be checked that \(\{d_{cut}(U,U^{(n)})\}_{n}\) converges to zero. Therefore \(U\) as the cut-distance convergent limit of the sequence \(\{U^{(n)}\}_{n \ge 1}\) of bounded graphons is an unbounded graphon.

Renormalization. We apply a new regulator \(z \in {\Delta}^{\ast}\) to define bounded regulated graphons together with related integrals \[U_{z}(x,y) := \sum\limits_{n=1}^{\infty} \frac{n}{1+|z|n} {1}_{I_{n} \times I_{n}}(x,y) \ , \ T(z):=\int_{[0,1]^{2}} U_{z}(x,y) dxdy, \tag{61}\] such that \(\{T(z)\}_{z}\) might diverges when the regulator \(z\) tends to zero, however, these divergences are isolated as poles in \(z\). The minimal subtraction map \(R_{ms}\) removes the divergent part to get \[T_{ren} = {lim}_{z \rightarrow 0}\bigg(T(z) – {divergent \ part}\bigg). \tag{62}\]

Consider the regulator \(z\) as a deformation parameter, the regulated graphons \(U_{z}\) as deformed versions of \(U\) where singularities are untangled, and the minimal subtraction map \(R_{ms}\) as a machinery for the removal of the divergent part of the formal series in \(z\) which is the same as the subtracting singular terms in a star product expansion. This dictionary leads us to formulate a new family of deformed star products on the algebra \(\mathcal{H}^{\mathfrak{U}}\) given by \[\label{z-star-1} W_{1} \star_{z} W_{2}:= W_{1}W_{2} + |z|B_{1}(W_{1},W_{2}) + |z|^{2}B_{2}(W_{1},W_{2}), \tag{63}\] such that the bi-differential operators \(B_{n}\), as the analogs of their classical ones, are definable on the basis of homomorphism densities \({ hom}(G,W_{i})\) of finite graphs with respect to graphons \(W_{1},W_{2}\). The renormalized graphon \(U_{ren}\) is well-defined by subtraction of divergent terms in the \(z\)-expansion of \(U_{z} \star_{z} U_{z}\).

The operator \(N_{U}=\sum\limits_{n=1}^{\infty}\frac{1}{n}W_{n}\) is a Nijenhuis map on the free commutative graded finite type algebra \(\mathcal{H}^{\mathfrak{U}}\) such that \(N_{U}(W_{n})=\frac{1}{n}W_{n}\), \(N_{U}(W_{i_{1}}…W_{i_{k}}) = \bigg(\sum\limits_{j=1}^{k} \frac{1}{j} \bigg)W_{i_{1}}…W_{i_{k}}\) and for any monomial \(f(V_{i_{1}},…,V_{i_{k}})\) of graphons, \(N_{U}(f)=\sum\limits_{n=1}^{\infty} \frac{1}{n}W_{n}f\). It defines a new star product \[W_{1} \star_{N_{U}} W_{2} = W_{1}W_{2} + |z| \{W_{1},W_{2}\}_{N_{U}} + |z|^{2} B_{2}(W_{1},W_{2}) + …, \tag{64}\] such that the Poisson bracket \(\{W_{1},W_{2}\}_{N_{U}}:=\{N_{U}(W_{1}),W_{2}\}_{{K},U} + \{W_{1},N_{U}(W_{2})\}_{{K},U}\) is a deformation of the Poisson bracket \[\begin{aligned} \{W_{1},W_{2}\}_{{K},U}(W):=&\{F_{W_{1}},F_{W_{2}}\}_{{K},U}(W)\notag\\ =& \frac{1}{2}\int_{[0,1]^{4}} \left(\frac{\delta F_{W_{1}}}{\delta W}(x,y,W){K}U(u,v)\frac{\delta F_{W_{2}}}{\delta W}(u,v,W)\right.\notag\\ &\left. – \frac{\delta F_{W_{2}}}{\delta W}(x,y,W){K}U(u,v)\frac{\delta F_{W_{1}}}{\delta W}(u,v,W) \right)dxdydudv, \end{aligned} \tag{65}\] with respect to some suitable kernel \({K}=K(x,y,u,v)\) such that \(F_{W_{i}}\) is the operator on the Hilbert space \(L^{2}(\mathcal{W}^{\mathfrak{U}}_{\approx}(\mathbb{R}))\) corresponding to \(W_{i}\) given by \[f \mapsto F_{W_{i}}(f):= \int_{\mathcal{W}^{\mathfrak{U}}_{\approx}(\mathbb{R})} W_{i}(x_{1},x_{2}) f(x_{2}) d\mu_{Borel}(x_{2}) \ . \tag{66}\]

Here the space of graphons is considered as a metric space with its corresponding Borel \(\sigma\)-algebra or a subset of a \(L^{p}\)-space with its Borel \(\sigma\)-algebra. The limit of the sequence \(\{U^{(n)} \star_{N_{U}} U^{(n)}\}_{n}\) recovers the renormalization of \(U\). It is enough to add some bounded graphons \(W_{-1},W_{0}\) to \(U\) to define some new regulated graphons \(V_{z} := |z|^{-1} W_{-1} + W_{0} + |z|W_{1} + …\) such that \(R_{ms}(V_{z}) = W_{0} + |z|W_{1}+…\). We get the renormalized version of the deformed star product (63) given by \[V_{ren} \tilde{\star}_{z} V_{ren}:= R_{ms}(V_{|z|}V_{|z|}) + |z| R_{ms}(B_{1}(V_{|z|},V_{|z|})) + |z|^{2}R_{ms}(B_{2}(V_{|z|},V_{|z|})) + … \ . \tag{67}\]

Applying some regularized character \(\phi^{z}\), such as the one given by (44) together with the Hopf–Birkhoff factorization \(\phi^{z} = (\phi_{-}^{z})^{\ast_{core}-1} \ast_{core} \phi_{+}^{z}\), we get \[\phi^{z}_{-} = |z|^{-1} W_{-1} \ , \ \phi^{z}_{+} = V_{ren}= R_{ms}(V_{z}) \ . \tag{68}\]

Analytic continuation. For the sequence \(\{W_{n}\}_{n \ge 1}\) which contributes to the infinite direct sum \(U\), we consider an approximating sequence \(\{Z_{s}\}_{s \ge 1}\) of graphons with nested supports given by \[Z_{s}(x,y) = {1}_{[0,r_{s}]^{2}}(x,y) \ \ , \ \ r_{s}=1-\frac{1}{s+1}, \tag{69}\] which converges to \({1}_{[0,1]^{2}}\). They are nested graphons such that (i) each \(Z_{s}\) can be approximated by some \(W_{k}\), \(k < s\), (ii) any measurable subgraphon of \(Z_{s}\) is a restriction of a smaller square, and (iii) the core coproduct (22) in Theorem 1 on these nested graphons are given by \[\Delta(Z_{s})=\sum\limits_{l=1}^{s} Z_{l} \otimes Z_{s-l} \ . \tag{70}\]

Replace \(N_{U}\) by the new Nijenhuis map \(N_{\tilde{U}}=\sum\limits_{n=1}^{\infty}\frac{1}{n}Z_{n}\) which is an almost everywhere bounded smooth map on \([0,1]^{2}\) and it increases when \((x,y)\) tends to \((0,0)\). In addition, \(N_{\tilde{U}}\) obeys the co-Nijenhuis identity \[\Delta(N_{\tilde{U}}(W)) = \bigg(N_{\tilde{U}} \otimes {id} +{id} \otimes N_{\tilde{U}} – N_{\tilde{U}} \otimes N_{\tilde{U}}\bigg)(\Delta(W)), \tag{71}\] with respect to the core coproduct(22). This allows us to lift \(N_{\tilde{U}}\) onto the dual space of the free commutative graded finite type topological Hopf algebra \(\mathcal{H}^{\mathfrak{U},cut}\). In other words, \(N_{\tilde{U}}\) induces a dual map \(N^{*}_{\tilde{U}}:L(\mathcal{H}^{\mathfrak{U},cut},A_{dr}) \rightarrow L(\mathcal{H}^{\mathfrak{U},cut},A_{dr})\) given by \(N^{*}_{\tilde{U}}(\phi)(W)=\phi(N_{\tilde{U}}(W))\) which obeys the Nijenhuis identity.

Define the noncommutative associative algebra \(C_{N_{\tilde{U}}}:=(L(\mathcal{H}^{\mathfrak{U},cut},A_{dr}),\circ_{N_{\tilde{U}}})\) such that the multiplication \(\circ_{N_{\tilde{U}}}\) is a deformation of the convolution product \(\ast_{core}\) with respect to the Nijenhuis map \(N_{\tilde{U}}\). Thanks to Corollary 6, the noncommutative symplectic geometry built on the space of Hamiltonian derivations on \(C_{N_{\tilde{U}}}\) determines the star product \(\star_{\{.,.\}_{N_{\tilde{U}},z}}\) given by \[\phi \star_{\{.,.\}_{N_{\tilde{U}}},z} \psi = \phi \ast_{core} \psi + |z|\{\phi,\psi\}_{N_{\tilde{U}}} + O(|z|^{2}), \tag{72}\] such that, by applying Theorem 2, the continuum limit of \(\bigg\{\phi^{z}_{-} *_{core} \phi^{z} + |z|\{\phi^{z}_{-},\phi^{z}\}_{N_{\tilde{U}}} + O(|z|^{2}) \bigg\}_{z \rightarrow 0}\) describes the analytic continuation for \(\phi^{z}_{+}(\tilde{U})\) in \({\Delta}^{\ast}\). ◻

Proof. The sequence \(\{W_{n}\}_{n \ge 1}\), \(W_{n}(x,y)=n.{1}_{[0,1/n]^{2}}(x,y)\) of graphons represents a concentration into a shrinking infinitesimal square around the origin in the plane. Each \(W_{n}\) has the total integral \(1\) such that by increasing \(n\) the local height inside the infinitesimal square grows but the sequence pointwise tends to zero almost everywhere. For large enough \(n\), each \(W_{n}\) can determine finite graphs with many edges concentrated in an infinitesimal region such that amplitudes generated by these concentrated substructures restore extremely short-distance interactions. In this scenario, if we translate these graphons to Feynman graphons, the corresponding Feynman amplitudes will have ultraviolet divergences. It is possible to assign Laurent series with finite pole parts to these amplitudes in terms of involving a short-distance regulator \(z\).

Rescaling or stretching methods allow us to determine a non-trivial unbounded graphon \(W\) defined on \([0,\infty)^{2}\) as the graph limit of the sequence \(\{W_{n}\}_{n \ge 1}\) when \(n\) tends to infinity. Define a regulated weight \(\omega_{z}(W)\) such as \(\omega_{z}(W) = c_{-1}(W)z^{-1} + c_{0}(W) + O(z)\) for \(W\). Here \(c_{-1}(W)\), as the coefficient of the simple pole in the regulator expansion of the regularized amplitudes of \(W\), represents the ultraviolet divergent strength. In addition, \(c_{0}(W)\), as the finite pole part, is the renormalizable finite amplitude which remains after subtracting divergences.

Thanks to Theorem 2, the renormalization of \(W\) with respect to the regularized character \(\phi\) is given by the recursive relations (47), (48) and (49) such that \[\phi_{-}^{z}(W) = – R_{ms} \bigg(\omega_{z}(W) + \sum \phi^{z}_{-}(W') \omega_{z}(W'') \bigg) \ , \ R_{ms}(\omega_{z}(W)) = c_{-1}(W)z^{-1}, \tag{73}\] with respect to the Sweedler notation for coproducts.

The Nijenhuis operator \(N_{1}= 2 R_{ms} – {Id}\) generates a finite weight such that after canceling of \(z^{-1}\) terms via the Nijenhuis identities, this finite weight equals the finite coefficient \(c_{0}(W)\) up to \(O(z)\). If we replace singular coefficients \(\omega_{z}(W)\) by \(c_{0}(W)\), then we get the renormalized star product \[f \star_{N_{1},ren,W} g = {exp} \bigg(\sum\limits_{\tilde{W}}c_{0}(\tilde{W})B_{\tilde{W}}\bigg)(f,g), \tag{74}\] such that the sum is over the chosen decomposition of the graphon \(W\) into its building blocks called graphon-modes.

\(\bullet\) For any finite graph \(G\), \(B_{G}\) is a bi-differential operator defined by assigning one derivative to each edge in terms of the combinatorial information of the graph and here \(B_{\tilde{W}}\), as the natural continuum analogue, are bi-differential operators associated to the graphon-modes of \(W\).

\(\bullet\) If \(\tilde{W}\) is the indicator of a block mode, then \(B_{\tilde{W}}\) is the block-weighted sum of the corresponding finite-graph bi-differential operators restricted to that region.

\(\bullet\) If \(W\) concentrates near a point, then \(B_{\tilde{W}}\) approximates a local bi-differential operator which generates ultraviolet divergences removable by renormalization.

\(\bullet\) The renormalized star product is independent of the choice of the graphon-mode decompositions because different decompositions only affects the intermediate coefficients and operators.

\(\bullet\) It is possible to assign different renormalized start products to the renormalization of \(W\) in terms of changing the Nijenhuis operator \(N_{1}\) by \(N_{\lambda}\). ◻

Proof. Each \(W_{n}\), as a step graphon, is constant on each block equal to the mean of \(W\) over that block. Consider the single graphon-mode \({ 1}_{[0,\alpha]^{2}}\) and the regulated weight \(\omega_{z}^{(n)}=A_{n}z^{-1} + B_{n} + O(z)\) such that sequences \(\{A_{n}\}_{n},\{B_{n}\}_{n}\) converge respectively to \(A,B\). The minimal subtraction map generates the counterterm \(-A_{n}z^{-1}\) where its corresponding renormalized coefficient \(c_{0,n}^{ ren}=B_{n}\) tends to \(B\) and it defines the limit weight for \(W\). For the Poisson tensor \(\pi\), consider the Nijenhuis operator \(N_{1}= 2 R_{ms} – {Id}\) together with the bi-differential operator \[B_{W}(f,g)(x)= (a.{1}_{[0,\alpha)}(x))\pi^{ij}(x) \partial_{i}f(x)\partial_{j}g(x), \tag{75}\] to define \[f \star_{N_{1},{ren},n} g = {exp} \bigg( c_{0,n}^{ren}B_{W_{n}} \bigg)(f,g) \ \rightarrow \ f \star_{N_{1},{ren},W} g = {exp} \bigg(BB_{W} \bigg)(f,g), \tag{76}\] such that the renormalized star product \(\star_{N_{1},{ren},W}\) can be described as the limit of the renormalized star products built from the initial approximating sequence and Nijenhuis operator \(N_{1}\). ◻

Proof. The sequence \(\{V_{n}\}_{n \ge 1}\) is convergent to \(V(x,y)=\frac{1}{xy}\) when \(n\) tends to infinity. For \(n \ge 1\), the bounded operator \[(N_{n}f)(x) = \int_{[0,1]} V_{n}(x,y) f(y) dy, \tag{77}\] is a Nijenhuis operator on a suitable subalgebra of \(L^{2}([0,1])\) and it defines the star product \[f \star_{N_{n}} g = N_{n}(f)g + f N_{n}(g) – N_{n}(fg) \ . \tag{78}\]

The cut-distance convergence implies the limiting \(N_{n} \rightarrow N\) with respect to the weak operator topology such that the renormalized Nijenhuis operator \[(N_{ren}f)(x) = {Finite-Part} \int_{[0,1]} \frac{f(y)}{xy} dy, \tag{79}\] is extracted by minimal subtraction map. ◻

Proof. For any \(\epsilon > 0\), define the bounded bilinear integral operator \[N_{\epsilon,V}(x,y):= \int_{[0,1]^{2}} \frac{V(u,v)}{(x+\epsilon)(y+\epsilon)(u+\epsilon)(v+\epsilon)} dudv, \tag{80}\] which acts on \(\mathcal{H}^{\mathfrak{U},{cut}}\). These operators regulate singularities at \(x,y,u,v=0\). When \(\epsilon\) tends to zero, the kernel \(K_{\epsilon}(x,y,u,v)=\frac{1}{(x+\epsilon)(y+\epsilon)(u+\epsilon)(v+\epsilon)}\) has a Laurent series in \(\epsilon\) with some divergent terms. Minimal subtraction map removes pole terms to generate the renormalized Nijenhuis operator \[N_{{ren},V}(x,y) = {Finite-Part} \int_{[0,1]^{2}} \frac{V(u,v)}{xyuv} dudv \ . \tag{81}\]

The renormalized star product \[W_{1} \star_{N_{{ren},V}} W_{2} = N_{{ren},V}(W_{1})W_{2} + W_{1}N_{{ren},V}(W_{2}) – N_{{ren},V}(W_{1}W_{2}), \tag{82}\] encodes the renormalization underlying the limiting \(N_{\epsilon,V} \rightarrow N_{{ren},V}\). ◻

Proof. It is a consequence of Lemmas 2, 3, Theorem 1 and Corollaries 4, 6 where \(X=\sum\limits_{n=0}^{\infty}c^{n}X_{n}\) is given by the recursive relations \(X_{n} = \sum\limits_{j=1}^{n} B_{\bullet_{n}}^{+}\bigg(\sum\limits_{k_{1}+…+k_{j+1}=n-j,k_{i}\ge 0} X_{k_{1}}…X_{k_{j+1}} \bigg)\) with \(X_{0}=\mathbb{I}\).

Thanks to [13, 22, 26, 27], the collection \(\{X_{n}\}_{n \ge 1}\) generates a graded Hopf subalgebra \(H_{E}\) of \(H_{CK}(\mathbb{N})\) which embeds in \(\mathcal{H}^{\mathfrak{U},{cut}}\) such that \({char}_{\lambda} H_{E} \subset C_{\lambda,{E}}= (L(H_{E},A_{{dr}}),\circ_{\lambda})\). The injective morphism \(H_{E} \rightarrow \mathcal{H}^{\mathfrak{U},{cut}}\) of Hopf algebras defines a surjective morphism \({char}_{\lambda} \mathcal{H}^{\mathfrak{U},{ cut}} \rightarrow {char}_{\lambda} H_{E}\). It allows us to lift the symplectic geometry model introduced in Corollary 6 onto \(C_{\lambda,{E}}\). ◻