Assign a nonnegative integer range \(r(X)\) to each support \(X\subseteq V\). Let \(X_0,\ldots ,X_m\) satisfy \(X_j\cap X_{j+1}\ne \varnothing \), and suppose any two sites of \(X_j\) can be joined by a walk in \(G\) of length at most \(r(X_j)\). If \(x\in X_0\) and \(y\in X_m\), there is a walk from \(x\) to \(y\) of length at most \(\sum _{j=0}^m r(X_j)\). For the common range \(R\), this is \((m+1)R\).

Under the common-range hypotheses of Theorem 36.1.1, \(d_G(x,y)\le (m+1)R\), and \(x,y\) belong to the same connected component. Consequently no such chain exists when the components differ, when \(d_G(x,y)=\infty \), or when \((m+1)R{\lt}d_G(x,y)\).

The function \(\ell \) satisfies the triangle inequality. Let \(\iota :V\to \mathbb {Z}^2\) assign integer coordinates such that each graph edge has displacement at most one. A walk \(p\) from \(a\) to \(b\) then satisfies \(\ell (\iota (a),\iota (b))\le |p|\). If \(d_G(a,b)\le R{\lt}\infty \), the displacement is at most \(R\).

Let \(B\) be a finite neighborhood and let \(\mathcal F\) be a finite family of distinct subsets of \(B\), each containing \(a\). Then \(|\mathcal F|\le 2^{|B\setminus \{ a\} |}\). If \(a\in B\), this equals \(2^{|B|-1}\).

Let \(\mathcal F\) be a finite family of distinct supports, with weights \(w_X\le J\) and \(J\ge 0\). Suppose each support containing \(a\) is contained in \(B\), where \(a\in B\). Then \(\sum _{X\in \mathcal F:\, a\in X}w_X\le 2^{|B|-1}J\).

For nonnegative weights on a finite family of distinct supports and any finite region \(S\),

If each inner site sum is at most \(b\), the left side of (1) is at most \(|S|b\).

Suppose the original weights are nonnegative, every original support has cardinality at most \(v\), and every site budget is at most \(b\ge 0\). Then \(W_n(X)\le (vb)^n\) for each \(X\in \mathcal F\).

The diamond \(D_R(a)\) has cardinality \(v_R=1+2R(R+1)\). Let \(\iota :V\to \mathbb {Z}^2\) be injective. A finite set \(S\subseteq V\) with \(\ell (\iota (a),\iota (x))\le R\) for each \(x\in S\) has at most \(v_R\) elements. If every graph edge has displacement at most one, the same bound holds when every \(x\in S\) is reached from \(a\) by a graph walk of length at most \(R\), or when \(d_G(a,x)\le R\). This is the local counting bound of [ 247 , Section 2 ] .

Let \(V\) be a finite graph embedded injectively in \(\mathbb {Z}^2\), with edge displacement at most one. Let \(\mathcal F\) be a finite family of distinct finite supports such that any two sites in a support admit a walk in the full graph of length at most the nonnegative integer \(R\). Put \(v_R=1+2R(R+1)\) and \(\mu _R=2^{v_R-1}\). Each original support has at most \(v_R\) sites, and at most \(\mu _R\) original supports contain any specified site. For \(J\ge 0\) and real weights \(w_X\le J\),

If also \(w_X\ge 0\) for each \(X\in \mathcal F\), then

The continuation bound holds for each original support \(X\) and permits repetitions. Both bounds apply to \(w_X=\lVert h_X\rVert \) for elements \(h_X\) of any seminormed additive group with \(\lVert h_X\rVert \le J\). These are the interaction budgets used in [ 247 , Sections 2 and 4 ] .

If \(w_Y\ge 0\) for each \(Y\in \mathcal F\), then \(0\le T_n^y(X)\le W_n(X)\) for every finite support \(X\) and every \(n\ge 0\).

Suppose every pair of sites in each \(X\in \mathcal F\) admits a walk of length at most the nonnegative integer \(R\). Fix \(X\in \mathcal F\) and an anchor \(a\in X\). For arbitrary real weights, \(T_n^y(X)=0\) whenever there is no walk from \(a\) to \(y\) of length at most \((n+1)R\). In particular, it vanishes if \((n+1)R{\lt}d_G(a,y)\), including \(d_G(a,y)=\infty \).

Under the range hypotheses of Theorem 36.5.3, suppose \(w_Y\ge 0\), \(|Y|\le v\) for each \(Y\in \mathcal F\), and \(\sum _{Y\in \mathcal F:\, x\in Y}w_Y\le b\) for every site \(x\), where \(v\) is a nonnegative integer and \(b\ge 0\). For \(X\in \mathcal F\), \(a\in X\), and \(n\ge 0\),

In the finite lattice setting of Theorem 36.4.3, one may take \(v=v_R\) and \(b=\mu _RJ\).

Under the hypotheses of Theorem 36.5.4, let \(d_G(a,y)=D\) for a nonnegative integer \(D\), and let \(\mu \ge 0\). Put \(\kappa =vb\). Then

Under the hypotheses of Theorem 36.6.1, the series in (11) converges and satisfies

This bounds the scalar interaction-chain series used in the proof of [ 247 , Lemma 4.1 ] .

Under the range hypotheses of Theorem 36.5.3, if \(d_G(a,y)=\infty \), then \(S_t^y(X)=0\) for every real \(t\) and arbitrary real weights, including signed weights.

For any two elements \(X,A\) of a normed ring, \(\| XA-AX\| \leq 2\| A\| \| X\| \).

Let \(\mathcal A\) be a complete real normed star algebra satisfying the \(C^*\) identity, and let \(K^*=-K\). For \(A,B\in \mathcal A\) and real \(t\),

Fix integers \(q\geq 1\) and \(R\geq 0\), and a real number \(J\). There are positive constants \(C,v,c\) with the following property. Let \(\Lambda \subseteq \mathbb {Z}^2\) be any finite domain, and let \(H\) be a finite-range Hamiltonian of range \(R\) and term-norm bound \(J\) on \(\Lambda \). For a region \(S\subseteq \Lambda \), write \(N_l(S)=\{ y\in \Lambda :d_\Lambda (y,S)\leq l\} \), with \(N_l(\varnothing )=\varnothing \). If \(A\) is supported on \(S\), then for every integer \(l\geq 0\) and real \(t\),

Empty domains and empty regions are included. This is the operator localization estimate used in [ 248 ] .

Fix integers \(q\geq 1\) and \(R\geq 0\), and a real number \(J\). There are positive constants \(C,v,c\) with the following property. Let \(\Lambda \subseteq \mathbb {Z}^2\) be a finite domain, let \(H'\) be a Hamiltonian of range \(R\) and term-norm bound \(J\) on \(\Lambda \), and let \(H\) be any Hermitian operator on the same space. Suppose that \(W\) acts on \(Q\subseteq \Lambda \) and that \(D=H'-H\) acts on \(Z\subseteq \Lambda \). For each integer \(l\geq 0\), put \(B=N_l(Q\cup Z)\), where distance is measured in the induced graph. There is a differentiable path \(U\) such that

For every real \(t\), the operator \(U(t)\) is unitary and acts on \(B\). Choose this path before the Gaussian variance and time cutoff. For every \(\eta \geq 0\) and \(T\geq 0\), let \(\gamma _\eta \) be the centered Gaussian measure of variance \(\eta \), and define

Then \(M_{\eta ,T}\) acts on \(B\), its norm is at most \(\| W\| \), and

The integrals retain their original Gaussian mass. Zero variance, empty domains and empty regions are included. This is the supported Gaussian approximation used in [ 248 ] .

Fix \(\sigma \in \Sigma \). If all factors in \(\prod _{v\in V}A_v((\tau _e)_{e\ni v},\sigma (v))\) are nonzero, then \(\tau _e=\sigma \) for every \(e\in T\). Conversely, this virtual assignment satisfies all equality constraints.

For every finite connected graph with \(|V|\ge 2\) and every \(\psi \colon \Sigma \to \mathbb {C}\), there are a spanning tree \(T\subseteq G\) and tensors with dimensions \(D_e=N\) on \(T\) and \(D_e=1\) off \(T\) such that

Every bond is positive and at most \(q^{|V|}\). If \(\psi \ne 0\), then \(c_A\ne 0\).

A connected graph with one vertex represents every coefficient function by its single physical tensor. Thus every finite connected graph admits the exact representation with positive bonds bounded by \(q^{|V|}\). For an empty vertex set, however, the contraction is the scalar one for every tensor family.

The open \(w\times h\) nearest-neighbor grid is the Cartesian product of the path graphs on \(w\) and \(h\) vertices. It is connected exactly when \(w,h{\gt}0\). If \(L\ge 2\), its \(L\times L\) vertex set has at least two elements.

For \(q,L\ge 1\), every vector \(\psi \) on the open \(L\times L\) square grid has an exact PEPS with \(1\le D_e\le q^{L^2}\). For \(L\ge 2\), the dimensions can be chosen as \(q^{L^2}\) on a spanning tree and one off that tree. If \(\| \psi \| =1\), its contraction \(\Phi \) satisfies \(\Phi =\psi \), \(\| \Phi \| =1\), and \(\| \| \Phi \| ^{-1}\Phi -1\cdot \psi \| =0\).

Fix \(q\ge 1\) and \(L_0\in \mathbb {N}\). For every \(0{\lt}L\le L_0\), every vector on the open \(L\times L\) square grid has an exact PEPS with positive bonds at most \(C_0=q^{L_0^2}\). This bound is independent of the vector and any Hamiltonian. For any \(C\in \mathbb {R}\) and \(r\ge 0\), the enlarged prefactor \(\max \{ C,C_0\} \) gives \(D_e\le \max \{ C,C_0\} L^r\) throughout the same range. If \(\chi {\gt}0\), then \(C_\chi =\max \{ C,C_0^{1/\chi }\} \) gives \(D_e\le (C_\chi L^r)^\chi \) on this range as well.

Fix integers \(q\ge 1\) and \(L_0\ge 0\), and real numbers \(C\) and \(c\ge 0\). Set \(C'=\max \{ C,q^{L_0^2}\} \). For every integer \(0{\lt}L\le L_0\) and every unit vector \(\Omega \) on the open \(L\times L\) square, there is a nonzero PEPS \(\Phi \) on that square with \(1\le D_e\le C'L^c\) on every edge and a real phase \(\theta \) such that \(\| \| \Phi \| ^{-1}\Phi -e^{i\theta }\Omega \| \le L^{-1}\). This is the finite-size part of  [ 248 , Section 8 ] (07-assembly.tex, lines 203–214).

Fix integers \(q\ge 1\) and \(L_0\ge 0\), and real numbers \(C\) and \(c\ge 0\). There is a constant \(C'{\gt}0\) with \(C'\ge C\) such that, for every integer \(2\le L{\lt}L_0\) and every unit vector \(\Omega \) on the open \(L\times L\) square, there is a nonzero PEPS \(\Phi \) on that square with \(1\le D_e\le C'L^c\) on every edge and a real phase \(\theta \) satisfying \(\| \| \Phi \| ^{-1}\Phi -e^{i\theta }\Omega \| \le L^{-1}\). The constant is chosen before \(L\) and \(\Omega \), as required for the finite-size argument in  [ 248 , Section 8 ] (07-assembly.tex, lines 203–214).

Let \(q,L\) be natural numbers, \(C,C',c\) real numbers with \(C\le C'\), and \(\Omega \) a vector on the open square. If \(\mathcal A_{C,c}(L,\Omega )\), then \(\mathcal A_{C',c}(L,\Omega )\).

Fix integers \(q\ge 1\) and \(L_0\ge 0\), real \(J,\Delta ,C\), and \(c\ge 0\). Suppose \(\mathcal A_{C,c}(L,\Omega )\) holds for every integer \(L\ge \max \{ 2,L_0\} \) and every \((H,E_0,\Omega )\in \mathcal G_{q,J,\Delta }(L)\). There is one \(C'{\gt}0\) with \(C'\ge C\) such that \(\mathcal A_{C',c}(L,\Omega )\) holds for every integer \(L\ge 2\) and every \((H,E_0,\Omega )\in \mathcal G_{q,J,\Delta }(L)\). The exponent \(c\) is unchanged, and \(C'\) is independent of \(L,H,E_0,\Omega \). This is the finite-size argument in  [ 248 , Section 8 ] (07-assembly.tex, lines 203–214).

The uniform polynomial approximation assertion

is equivalent to

In both assertions, the constants are chosen before the size, Hamiltonian, energy, and ground vector.

Let \(L\ge 1\), \(C\ge 0\) and \(c\le c'\). If \(\mathcal A_{C,c}(L,\Omega )\), then \(\mathcal A_{C,c'}(L,\Omega )\).

If the assertion of Definition 36.12.1.2 holds, then the uniform polynomial approximation assertion (27) holds: for every \(q\ge 2\) and \(J,\Delta {\gt}0\) there are \(C',c'{\gt}0\) such that \(\mathcal A_{C',c'}(L,\Omega )\) for every \(L\ge 2\) and every \((H,E_0,\Omega )\in \mathcal G_{q,J,\Delta }(L)\). This is the argument of  [ 248 , Section 8 ] (07-assembly.tex, lines 53–214) after the compressed network has been obtained.