Proof. Set \(c_f=e^{h_f}\). Orient the edges temporarily, delete one row of the oriented incidence matrix, and denote the resulting columns by \(a_f\). The reduced weighted Laplacian is \[L=\sum_{f\in E(H)}c_fa_fa_f^{\mathsf T}.\] Write \(r=|V(H)|-1\). Cauchy–Binet, applied to the reduced incidence matrix with column \(f\) multiplied by \(\sqrt{c_f}\), gives \[\det L =\sum_{\substack{F\subseteq E(H)\\|F|=r}} \det(a_f:f\in F)^2\prod_{f\in F}c_f =\sum_{T\in\mathcal T(H)}\prod_{f\in T}c_f =:\mathcal Z_H(h).\] For the second equality, a set containing a cycle has dependent incidence columns, while an acyclic set of \(r\) edges is a spanning tree. Its reduced incidence determinant has absolute value one: expand at a leaf other than the vertex whose row was deleted and continue by induction. Connectedness supplies a spanning tree, so \(\mathcal Z_H(h)>0\) and \(L\) is invertible.

Differentiating the finite partition sum gives both \(p_e=\partial_{h_e}\log\mathcal Z_H\) and the covariance identity in (1). Differentiating the determinant instead gives \[p_e=c_ea_e^{\mathsf T}L^{-1}a_e.\] Since \(\partial_{h_f}L^{-1}=-c_fL^{-1}a_fa_f^{\mathsf T}L^{-1}\), for distinct edges \(e,f\) we obtain \[ \operatorname{Cov}_h(\xi_e,\xi_f) =-c_ec_f\bigl(a_e^{\mathsf T}L^{-1}a_f\bigr)^2\le0. \tag{3}\] Every spanning tree has exactly \(r\) edges. Consequently \[\sum_{f\in E(H)}\operatorname{Cov}_h(\xi_e,\xi_f) =\operatorname{Cov}_h\!\left(\xi_e, \sum_{f\in E(H)}\xi_f\right)=0.\] The diagonal covariance is \(p_e(1-p_e)\); all the other covariances are nonpositive by (3). Their absolute sum is therefore \(2p_e(1-p_e)\le1/2\), as claimed. This also covers a bridge: its inclusion probability is one and its entire covariance row is zero. Finally, integrate the gradient along the segment from \(h\) to \(h'\) to obtain (2). The estimate holds at every point of that segment, without a restriction on the weights. ◻

Proof. For finite \(S\subset I\), write \(\mathcal H_t^S=\sigma\{Y_j(s):j\in S,\ 0\le s\le t\}\). When \(t>0\), the density of the endpoint vector \((Y_j(t))_{j\in S}\), conditional on \(X\), relative to independent \(N(0,t)\) variables is \[\exp\!\left(\sum_{j\in S}(y_j-t/2)X_j\right).\] Here \(X_j^2=X_j\). Moreover, \[Y_j(s)-\frac{s}{t}Y_j(t) =B_j(s)-\frac{s}{t}B_j(t),\qquad 0\le s\le t.\] These bridge paths are jointly independent of \(X\) and the endpoint vector. Indeed, at finitely many times this follows from zero Gaussian covariances and independence of \(B\) from \(X\); continuity and rational-time evaluation extend it to the paths. Bayes’ formula therefore gives \[b_{i,S}(t,Y(t))=\mathbb E[X_i\mid\mathcal H_t^S] \qquad\text{almost surely}.\] At \(t=0\) both sides equal \(\mathbb EX_i\).

For fixed \(i,t\), the sigma-fields \(\mathcal H_t^{S_n(i)}\) increase to generate \(\mathcal H_t\). The bounded conditional expectations \(M_n=\mathbb E[X_i\mid\mathcal H_t^{S_n(i)}]\) converge almost surely to \(\mathbb E[X_i\mid\mathcal H_t]\). We recall a short verification of this last step. For \(m\ge n\), \[\mathbb E|M_m-M_n|^2=\mathbb EM_m^2-\mathbb EM_n^2.\] Choose \(n_k\uparrow\infty\) so that this difference is at most \(2^{-4k}\) whenever \(m\ge n=n_k\). The maximal inequality obtained by conditioning on the first crossing gives \[\mathbb P\!\left(\max_{n_k\le r\le m}|M_r-M_{n_k}|>2^{-k}\right) \le 2^{2k}\mathbb E|M_m-M_{n_k}|^2\le2^{-2k}.\] To see the first inequality, split into the disjoint events of a first crossing at \(r\) and apply conditional Jensen to \(M_m-M_{n_k}\) at that stage. Letting \(m\to\infty\) and summing over \(k\) proves that the full sequence is almost surely Cauchy. Its bounded limit has the same integral as \(X_i\) against every event in each finite-stage sigma-field. A monotone-class argument identifies it with \(\mathbb E[X_i\mid\mathcal H_t]\). Thus the limsup defining \(b_i\) is an actual limit along the observations at this fixed time, and (13) follows. ◻

Proof. Fix \(T<\infty\). The map \((s,\omega)\mapsto Y(s,\omega)\) on \([0,T]\) is measurable for \(\mathcal B([0,T])\otimes\mathcal H_T\), coordinate by coordinate. Composing with the bounded Borel drift shows that the integral in (14) is defined, measurable from the observation history, and continuous in its upper endpoint. Hence \(\widehat W\) is coordinatewise continuous and adapted.

For \(0\le u<v\) and any bounded \(\mathcal H_u\)-measurable random variable \(K\), Lemma 9 gives, separately for each \(s\in[u,v]\), \[\mathbb E[Kb_i(s,Y(s))]=\mathbb E[KX_i].\] Bounded Fubini integrates these numerical identities. Since \(B_i(v)-B_i(u)\) is independent of \(X\) and the Brownian histories through \(u\), it is independent of \(\mathcal H_u\). Consequently \[ \mathbb E[\widehat W_i(v)-\widehat W_i(u)\mid\mathcal H_u]=0. \tag{15}\] This argument uses no single exceptional set valid for every real time.

We identify the joint law using conditional characteristic functions: the error on each small interval is of order \(h^{3/2}\), so the accumulated error vanishes under subdivision. Let \(\theta=(\theta_i)_{i\in I}\in\mathbb R^I\) have finite support, and put \[L_\theta=\sum_i|\theta_i|,\qquad \sigma_\theta^2=\sum_i\theta_i^2,\qquad h=v-u.\] The increment \[D_{u,v}=\sum_i\theta_i(\widehat W_i(v)-\widehat W_i(u))=Q+R\] has a conditionally Gaussian term \(Q=\sum_i\theta_i(B_i(v)-B_i(u))\) of conditional variance \(h\sigma_\theta^2\), and a correction satisfying \(|R|\le hL_\theta\). It follows, without assuming that \(Q\) and \(R\) are independent, that \[\begin{align*} \left|\mathbb E[D_{u,v}^2\mid\mathcal H_u]-h\sigma_\theta^2\right| &\le 2L_\theta\sigma_\theta h^{3/2}+L_\theta^2h^2,\\ \mathbb E[|D_{u,v}|^3\mid\mathcal H_u] &\le 4\bigl(c_3\sigma_\theta^3h^{3/2}+L_\theta^3h^3\bigr), \end{align*}\] where \(c_3=\mathbb E|N(0,1)|^3\). Together with (15), the Taylor bound \(|e^{\mathrm ix}-1-\mathrm ix+x^2/2|\le |x|^3/6\) yields a deterministic constant \(C_\theta\) such that, for \(h\le1\), \[ \left|\mathbb E[e^{\mathrm iD_{u,v}}\mid\mathcal H_u] -e^{-h\sigma_\theta^2/2}\right| \le C_\theta h^{3/2}. \tag{16}\]

Fix \(s<t\) and \(A\in\mathcal H_s\). Partition \([s,t]\) into \(N\) equal intervals with endpoints \(t_0=s,\ldots,t_N=t\) and length \(h\le1\). Set \[a=e^{-h\sigma_\theta^2/2},\qquad \mu_k=\mathbb E\!\left[\mathbf 1_A \prod_{r=1}^k e^{\mathrm iD_{t_{r-1},t_r}}\right].\] The preceding factors are \(\mathcal H_{t_{k-1}}\)-measurable and have modulus one. Thus (16) gives \(\mu_k=a\mu_{k-1}+\delta_k\) with \(|\delta_k|\le C_\theta h^{3/2}\). Iterating, and using \(0<a\le1\), gives \[\left|\mathbb E[\mathbf 1_Ae^{\mathrm iD_{s,t}}] -\mathbb P(A)e^{-(t-s)\sigma_\theta^2/2}\right| \le NC_\theta h^{3/2}\longrightarrow0.\] For every finite set of coordinates and every \(A\in\mathcal H_s\), uniqueness of finite-dimensional characteristic functions identifies the increment law restricted to \(A\) as \(\mathbb P(A)\) times the centered Gaussian law with covariance \((t-s)\operatorname{Id}\). This proves the vector increment law and independence from the entire observation past. Repeating over successive time intervals gives all joint finite-dimensional Brownian laws. Continuous paths, starting at zero, and rational-time evaluations determine the law on the countable product of continuous-path spaces. This proves the claim. ◻

Proof. All integrals below are ordinary scalar Lebesgue integrals. Define \[ Z^0(w)=w,\qquad Z_i^{k+1}(w)(t) =w_i(t)+\int_0^t b_i(s,Z^k(w)(s))\,ds, \qquad k\ge0. \tag{19}\] These iterates are defined for every input and have continuous coordinates. Indeed, a continuous coordinate path family gives a Borel map into \(\mathbb R^I\), so the integrand is Borel and bounded; its indefinite integral is continuous. The initial increment is bounded by \(t\), uniformly in \(i\). Induction using (17) gives \[ \sup_{i\in I}|Z_i^{k+1}(w)(t)-Z_i^k(w)(t)| \le \frac{L^k t^{k+1}}{(k+1)!},\qquad k\ge0, \tag{20}\] where \(L^0=1\), including when \(L=0\). At the induction step the differences of the two input fields are bounded by the preceding estimate, so the hypothesis of (17) applies.

For every finite \(T\), the bounds in (20), with \(t\) replaced by \(T\), have finite sum. Thus the iterates converge to a limit \(Z(w)\) uniformly over all coordinates and all \(t\in[0,T]\) in the sense that \[\sup_{i\in I}\sup_{0\le t\le T} |Z_i(w)(t)-Z_i^k(w)(t)|\longrightarrow0.\] Each coordinate of the limit is continuous. The Lipschitz estimate then gives uniform convergence of the corresponding drifts on \([0,T]\), allowing passage through each integral in (19). This proves (18). The uniform estimates concern differences from the same input: the input itself may be unbounded over \(I\), and its coordinates may have no common modulus of continuity.

For any other solution \(z\), boundedness of \(b\) gives \(\sup_i|z_i(t)-w_i(t)|\le t\). Repeating the same induction yields \[\sup_{i\in I}|z_i(t)-Z_i^k(w)(t)| \le \frac{L^k t^{k+1}}{(k+1)!}.\] Letting \(k\to\infty\) proves uniqueness for every input.

We next verify measurability on this entire path space. The evaluation map \((s,z)\mapsto z(s)\) from \([0,\infty)\times\mathcal C_I\) to \(\mathbb R^I\) is continuous. If \(Z^k\) is Borel, the integrand \((s,w)\mapsto b_i(s,Z^k(w)(s))\) is jointly Borel. Parameterized integration of a bounded Borel function shows that its integral from \(0\) to \(t\) is Borel in \((t,w)\). Consequently every rational-time coordinate evaluation of \(Z^{k+1}\) is Borel. Such evaluations generate the Borel sigma-field of \(\mathcal C_I\), proving inductively that every \(Z^k\) is Borel. Their limits at rational times prove that \(Z\) is Borel as well.

Finally, the iteration uses only times up to its time argument, which proves causality. A coordinate relabeling that intertwines the drifts also intertwines each step of (19) and hence its limit. This applies equally to a bijection between two different countable index sets with corresponding drifts. ◻

Proof. Each finite tilt is a ratio of finite sums with a strictly positive denominator, and is therefore jointly Borel in time and the field. Its limsup is jointly Borel and takes values in \([0,1]\). If \(d=\sup_j|y_j-y'_j|<\infty\), the assumed response bound gives \[|b_{i,S_n(i)}(t,y)-b_{i,S_n(i)}(t,y')|\le Ld\] for every \(i,n,t\). Taking limsups in both directions proves (17). Lemma 11 therefore applies. Formula (21) is a Borel predicate on every input, including those on which the displayed slopes do not converge.

To identify its full output law, use the reference observation model: take \(X\sim\nu\), independently take standard Brownian paths \((B_i)\), and put \(Y_i(t)=tX_i+B_i(t)\). Lemma 9 identifies the drift with the posterior mean at each fixed time, and Lemma 10 shows that \[\widehat W_i(t)=Y_i(t)-\int_0^t b_i(s,Y(s))\,ds\] has the law of independent standard Brownian paths. By this definition, the identity \[Y_i(t)=\widehat W_i(t)+\int_0^t b_i(s,Y(s))\,ds\] holds for all \(i\) and all \(t\) on every realization of the continuous paths. It does not require the fixed-time posterior identities to hold on one event simultaneously at every time. Deterministic uniqueness in Lemma 11 consequently gives \(Y=Z(\widehat W)\) as path families. Since \(Z\) is Borel, it follows that independent Brownian input \(w\) produces the complete joint path law of \(Y\) under \(Z\).

For each \(i\in I\) and \(\delta>0\), the Gaussian tail bound gives \[\mathbb P\bigl(|B_i(n)|>\delta n\bigr) \le 2\exp(-\delta^2n/2),\qquad n\ge1.\] These probabilities are summable. Borel–Cantelli and a countable intersection over \(i\) and positive rational \(\delta\) give \(B_i(n)/n\to0\) for every \(i\) simultaneously, almost surely. Hence \(Y_i(n)/n\to X_i\) simultaneously, and \(\Psi(\widehat W)=X\) almost surely. This identifies the entire configuration law of \(\Psi(w)\) as \(\nu\). Equivariance follows from Lemma 11 and the coordinatewise formula (21). ◻

Proof. The tilted generating polynomial is \[g_{\mu_h}(z) =\frac{g_\mu((e^{h_j}z_j)_{j\in F})} {g_\mu((e^{h_j})_{j\in F})}.\] The denominator is positive. Positive coordinate scaling preserves stability, so \(\mu_h\) is strongly Rayleigh; see Borcea, Brändén, and Liggett (Borcea et al. 2009, sec. 2.1(iv) and Proposition 3.1(2)). Differentiating the finite partition sum gives the covariance identity in (22).

Write \(n=|F|\) and \(\eta=\mu_h\). To recover a constant total count, take a disjoint set \(F'\) of \(n\) dummy coordinates. First sample the occupied set \(A\subseteq F\) with law \(\eta\); conditional on \(A\), occupy a uniformly chosen subset of \(F'\) of size \(n-|A|\). The resulting law \(\widetilde\eta\) keeps \(\eta\) as its \(F\)-marginal and has exactly \(n\) occupied coordinates in total. Write \(\eta(A)\) for the probability of the occupied set \(A\), and let \(e_k\) be the elementary symmetric polynomial of degree \(k\) in \(w=(w_j)_{j\in F'}\). The generating polynomial of \(\widetilde\eta\), called the symmetric homogenization of \(g_\eta\), is \[\widetilde g(z,w) =\sum_{A\subseteq F}\eta(A)\prod_{j\in A}z_j\, \frac{e_{n-|A|}(w)}{\binom{n}{|A|}}.\] The polynomial has nonnegative coefficients, equals one at the all-one vector, is homogeneous of degree \(n\), and specializes to \(g_\eta\) when all the \(w\) variables equal one. The additional fact that it is stable is supplied by (Borcea et al. 2009, Definition 2.12 and Theorem 4.2). Thus \(\widetilde\eta\) is strongly Rayleigh.

Let \(C_{jk}=\mathop{\mathrm{Cov}}_{\widetilde\eta}(X_j,X_k)\). The Rayleigh inequality in (Borcea et al. 2009, Theorem 4.1), evaluated at the all-one vector, gives \(C_{jk}\le0\) for \(j\ne k\). Since the total count is constant, for \(i\in F\) we have \[0=\sum_{j\in F\sqcup F'}C_{ij} =C_{ii}+\sum_{j\in F\setminus\{i\}}C_{ij} +\sum_{j\in F'}C_{ij}.\] The dummy covariances in the last sum are nonpositive. It follows that \[-\sum_{j\in F\setminus\{i\}}C_{ij} =C_{ii}+\sum_{j\in F'}C_{ij}\le C_{ii}, \qquad \sum_{j\in F}|C_{ij}|\le2C_{ii}.\] The covariances with both indices in \(F\) are those of \(\eta\), and \(C_{ii}=p_i(h)(1-p_i(h))\le1/4\). This proves (22), including degenerate coordinates. No homogeneity or full-support assumption on \(\mu\) was used. Integrating the gradient along the segment from \(h\) to \(h'\) proves (23). ◻

Proof of Theorem 3. Fix a site \(i\in\Gamma\) and a finite observed set \(S\subseteq\Gamma\). Let \(J=S\cup\{i\}\) and let \(\nu_J\) be the marginal of \(\nu\) on \(J\). For \(t\ge0\) and \(y\in\mathbb R^\Gamma\), tilt \(\nu_J\) by \[\exp\!\left(\sum_{j\in S}(y_j-t/2)x_j\right).\] The resulting probability law \(\eta_{t,y}\) is an arbitrary positive external-field tilt of the strongly Rayleigh law \(\nu_J\), with zero log field on \(J\setminus S\). Its mean at \(i\) is exactly the finite-tilt function \(b_{i,S}(t,y)\) in (7). Finite differentiation and Lemma 15 give \[\frac{\partial b_{i,S}}{\partial y_j}(t,y) =\mathop{\mathrm{Cov}}_{\eta_{t,y}}(X_i,X_j)\quad(j\in S), \qquad \sum_{j\in S}\left|\frac{\partial b_{i,S}}{\partial y_j}(t,y)\right| \le\sum_{j\in J}|\mathop{\mathrm{Cov}}_{\eta_{t,y}}(X_i,X_j)|\le\frac12.\] Integrating along a segment of fields yields \[ |b_{i,S}(t,y)-b_{i,S}(t,y')| \le\frac12\max_{j\in S}|y_j-y'_j|. \tag{24}\] For \(S=\varnothing\), the finite-tilt function is constant and the same statement holds with the empty maximum defined to be zero. The bound is uniform in \(i,S,t\) and in all finite field values. It uses the actual finite marginal of \(\nu\), with no infinite external-field tilt or finite-volume approximation.

Choose finite sets \(F_n\subseteq\Gamma\) increasing to \(\Gamma\), with the identity in every \(F_n\), and put \[S_n(h)=hF_n,\qquad h\in\Gamma.\] These are finite exhaustions at every site, and \(S_n(gh)=gS_n(h)\) for all \(g,h\in\Gamma\). This requires neither finite generation nor amenability; for a finite group one may take \(F_n=\Gamma\). Invariance of \(\nu\) gives the exact finite-tilt identity \[ b_{gh,gS}(t,\tau_g y)=b_{h,S}(t,y) \tag{25}\] for every \(g,h,t,y\) and finite \(S\). Indeed, in the numerator and denominator of (7), substitute \(x=\tau_g z\). Then \(x_{gh}=z_h\), \((\tau_g y)_{gs}=y_s\), and \(x_{gs}=z_s\), while the law of \(z\) is again \(\nu\). All sums are finite, so the identity holds pointwise even when the complete field is unbounded.

Define the limsup drift using these \(S_n(h)\). Equations (25) and \(S_n(gh)=gS_n(h)\) give \[b_{gh}(t,\tau_g y) =\limsup_n b_{gh,S_n(gh)}(t,\tau_g y) =\limsup_n b_{h,S_n(h)}(t,y)=b_h(t,y).\] Thus the drift is equivariant under every regular left translation on its entire domain. The uniform bound (24) supplies the hypothesis of Proposition 12 with \(L=1/2\). That proposition gives a Borel map \(\Psi\) from the product continuous-path space to \(\{0,1\}^{\Gamma}\), sending independent standard Brownian paths to \(\nu\) and commuting with every \(\tau_g\) on every input.

The proof of Lemma 14 constructs an everywhere-defined Borel map from one uniform variable to a standard Brownian path, with fixed defaults on its exceptional inputs. Applying this map coordinatewise sends independent uniform labels on \(\Gamma\) to independent Brownian paths and commutes with all translations. Its composition with \(\Psi\) is the required map \(\Phi_\nu\). ◻

Proof of Corollary 4. For a countable discrete set, every self-adjoint positive contraction defines a determinantal probability law; see Lyons (Lyons 2003, sec. 8). For a finite \(J\subseteq\Gamma\), let \(K_J\) be the compression of \(K\) to \(\ell^2(J)\). It is a Hermitian positive contraction. For every \(A\subseteq J\), the inclusion probability of \(A\) under the \(J\)-marginal is \[\mathbb P(A\subseteq X)=\det K_{A,A}=\det (K_J)_{A,A}.\] These inclusion probabilities determine the finite binary law by inclusion–exclusion. Hence that marginal is the determinantal law with kernel \(K_J\), and it is strongly Rayleigh by (Borcea et al. 2009, Proposition 3.5). This covers complex Hermitian, singular, and projection kernels; no trace-class condition on the operator on \(\ell^2(\Gamma)\) is needed. Theorem 3 now applies whenever the determinantal law is translation invariant.

If \(K\) commutes with the left regular representation, its matrix entries satisfy \(K_{gh,gk}=K_{h,k}\). All finite inclusion determinants are then translation invariant. Inclusion–exclusion and the uniqueness of a countable law from its cylinder probabilities give invariance of the determinantal law, proving the final assertion. ◻