--- title: Random Interlacements Percolation url: https://www.emergentmind.com/topics/percolation-of-random-interlacements type: topic --- # Random Interlacements Percolation Searching arXiv for recent and foundational papers on random interlacements percolation. Found relevant papers including the foundational work on level-set percolation for random interlacements and the Gaussian free field [1302.7024], sharpness and threshold equality for the vacant set [2308.07919], local uniqueness in the small-intensity regime [1206.6635], decoupling on \(G\times \mathbb{Z}\) [1010.1490], and recent sharp connectivity bounds [2504.02777]. Percolation of random interlacements concerns connectivity properties of random subsets of \(\mathbb{Z}^d\), \(d \ge 3\), generated by a Poisson cloud of doubly-infinite nearest-neighbor trajectories. In the continuous-time framework, the basic objects are not only the classical interlacement set \(\mathcal{I}^u\) and vacant set \(\mathcal{V}^u\) at level \(u\), but also occupation-time level sets obtained by thresholding the local-time field \(L_{x,u}\). This yields a two-parameter family of occupied and empty phases whose percolative behavior can be analyzed through an exact isomorphism with the Gaussian free field, together with renormalization and decoupling methods [1302.7024]. ## 1. Continuous-time interlacements and occupation-time level sets Continuous-time random interlacements on \(\mathbb{Z}^d\), \(d \ge 3\), are defined as a Poisson point process on the space of doubly-infinite nearest-neighbor trajectories modulo time-shift, endowed with i.i.d. exponential holding times of parameter \(1\) on each discrete step. The intensity measure is built from potential theory through the equilibrium measure \(e_K\) and the capacity \(\mathrm{cap}(K)\) of finite sets. At level \(u \ge 0\), the interlacement set is \[ \mathcal{I}^u(\omega)=\bigcup_{u_i\le u}\mathrm{range}(X(\widehat w_i)), \] and the vacant set is \[ \mathcal{V}^u=\mathbb{Z}^d\setminus \mathcal{I}^u. \] The occupation-time field is \[ L_{x,u}(\omega)=\sum_{i\ge 0}\sum_{n\in\mathbb{Z}} \sigma_n(\widehat w_i)\mathbf{1}\{X_n(\widehat w_i)=x,\ u_i\le u\}, \] the total continuous time spent at \(x\) by all interlacement trajectories with label at most \(u\) [1302.7024]. Thresholding \(L_{x,u}\) produces the occupied and empty level sets \[ \mathcal{I}^{u,\alpha}=\{x\in\mathbb{Z}^d:L_{x,u}>\alpha\},\qquad \mathcal{V}^{u,\alpha}=\{x\in\mathbb{Z}^d:L_{x,u}\le \alpha\}, \] for \(\alpha\ge 0\). The classical model is recovered at \(\alpha=0\): \[ \mathcal{I}^u=\mathcal{I}^{u,0}=\{x:L_{x,u}>0\},\qquad \mathcal{V}^u=\mathcal{V}^{u,0}. \] Thus \(\mathcal{I}^{u,\alpha}\subset \mathcal{I}^u\) for all \(\alpha\ge 0\), whereas \(\mathcal{V}^{u,\alpha}\supset \mathcal{V}^u\). Increasing \(\alpha\) shrinks the occupied phase and enlarges the empty phase. The geometric interpretation is that time-thresholding removes lightly visited sites from the interlacement cluster and can therefore destroy long-range connectivity even though the underlying trajectory cloud remains extensive. ## 2. Gaussian free field isomorphism and level-set translation A central structural input is Sznitman’s isomorphism theorem. Let \((\varphi_x)_{x\in\mathbb{Z}^d}\) be the centered Gaussian free field with covariance \[ E[\varphi_x\varphi_y]=g(x,y), \] where \(g\) is the Green function of simple random walk. Then for every \(u\ge 0\), \[ \big(L_{x,u}+\tfrac12\varphi_x^2\big)_{x\in\mathbb{Z}^d} \quad\text{under }P\otimes P^G \] has the same law as \[ \big(\tfrac12(\varphi_x+\sqrt{2u})^2\big)_{x\in\mathbb{Z}^d} \quad\text{under }P^G. \] Equivalently, for any finite \(K\subset \mathbb{Z}^d\), \[ \big(L_{x,u}+\tfrac12\varphi_x^2\big)_{x\in K} \overset{\text{law}}{=} \big(\tfrac12(\varphi_x+\sqrt{2u})^2\big)_{x\in K}. \] This is an exact identity in law, not an independent coupling, but it permits direct comparison of occupation-time exceedance events with Gaussian free field level events [1302.7024]. For the occupied phase, the identity yields \[ L_{x,u}>\alpha \;\Longrightarrow\; \frac{(\varphi_x+\sqrt{2u})^2}{2}>\alpha \;\Longrightarrow\; |\varphi_x|\ge \sqrt{2\alpha}-\sqrt{2u}. \] Accordingly, for \(\alpha\ge u\), the law \(Q_{u,\alpha}\) of \((\mathbf{1}\{L_{x,u}>\alpha\})_{x\in\mathbb{Z}^d}\) is stochastically dominated by the law of the two-sided Gaussian free field level set \[ \mathcal{L}^{\ge h}=\{x:|\varphi_x|\ge h\},\qquad h(u,\alpha)=\sqrt{2\alpha}-\sqrt{2u}. \] For the empty phase, the same identity shows that the constraint \(L_{x,u}\le \alpha\) can only occur if either \(\varphi_x\) is sufficiently negative to counteract the shift \(+\sqrt{2u}\), or \(|\varphi_x|\) is itself large. This leads to upper bounds on connectivity in \(\mathcal{V}^{u,\alpha}\) when \(u\) is large relative to \(\alpha\). The Gaussian free field comparison relies on two classes of level sets. The two-sided level set is \[ \mathcal{L}^{\ge h}=\{x:|\varphi_x|\ge h\}, \] with critical threshold \[ \mathfrak{h}_*=\inf\{h\ge 0:P^G[0\leftrightarrow\infty\text{ in }\mathcal{L}^{\ge h}]=0\}. \] One has \(\mathfrak{h}_*<\infty\) for all \(d\ge 3\), and for sufficiently large \(h\), \[ P^G[B(0,L)\leftrightarrow S(0,2L)\text{ in }\mathcal{L}^{\ge h}] \le c e^{-c' L^\rho}. \] The one-sided level set \[ E_\varphi^{\ge h}=\{x:\varphi_x\ge h\} \] has its own threshold \(h_*\), finite for all \(d\ge 3\) and strictly positive in high dimensions. ## 3. Renormalization, sparse trees, and decoupling The proof architecture is multiscale. One fixes scales \[ L_n=l_0^nL_0,\qquad \mathbb{L}_n=L_n\mathbb{Z}^d, \] and boxes \[ B_{n,x}=x+\big([0,L_n)\cap\mathbb{Z}\big)^d. \] A sparse embedding of a dyadic tree into \(\mathbb{Z}^d\) produces \(2^n\) well-separated descendant boxes inside a parent box. This geometry allows one to define recursively propagating bad events and to quantify their decay through scale [1302.7024]. For localized events \(A_{0,x}\) depending only on coordinates in \(B_{0,x}\), one introduces \[ p_n^A(h) = \sup_{x\in\mathbb{L}_n,\ \mathcal{T}\in\Lambda_{n,x}} P^G\Big[\bigcap_{\text{leaves }m\in T^{(n)}} \Phi_h^{-1}(A_{0,\mathcal{T}(m)})\Big], \] where \(\Phi_h((\varphi_x))=(\mathbf{1}\{\varphi_x\ge h\})\). The decoupling inequality has the form \[ p_{n+1}(h_{n+1}) \le p_n(h_n)^2+3\exp\{-(\beta_n-M(n,L_0))^2\}, \] with \(M(n,L_0)\asymp \sqrt{\log(2^nL_0^d)}\), and the sprinkling increment \(h_{n+1}-h_n\) proportional to \[ c\,r^{d-2}\big(2l_0^{-(d-2)}\big)^{n+1}\beta_n. \] Choosing \(\beta_n\) appropriately propagates the recursion and yields stretched-exponential bounds. The resulting exponent is \[ \rho=\frac{\log 2}{\log l_0}. \] The geometric counterpart is the cascading principle for bad blocks. One declares a level-\(0\) box bad when it contains a local obstruction, and defines a level-\(n\) bad event when \(2^n\) well-separated bad level-\(0\) boxes are embedded inside the corresponding \(L_n\)-box. If \[ P[\text{bad }A_{n,x}] \le 2^{-2^n} \] for all \(n\) and \(x\), then long connections through bad boxes satisfy the same doubly exponential scale estimate, which interpolates into the stretched-exponential annulus-crossing bounds used throughout the theory. This machinery is used both for Gaussian free field level sets and, via the isomorphism, for occupation-time level sets of interlacements. ## 4. Critical parameters for occupied and empty phases The occupation-time model introduces two natural critical families. For fixed \(u\), the occupied-set threshold is \[ \alpha_*(u)=\inf\{\alpha\ge 0: P[0\leftrightarrow\infty\text{ in }\mathcal{I}^{u,\alpha}]=0\}, \] and for fixed \(\alpha\), the empty-set threshold is \[ u_*(\alpha)=\inf\{u\ge 0: P[0\leftrightarrow\infty\text{ in }\mathcal{V}^{u,\alpha}]=0\}. \] Before the summary table, two structural facts are essential. First, for \(u>0\) and \(\alpha>0\), the law \(Q_{u,\alpha}\) of \((\mathbf{1}\{L_{x,u}>\alpha\})_{x\in\mathbb{Z}^d}\) is translation invariant and satisfies the finite energy property \[ 00\); non-decreasing in \(u\); \(\alpha_*(0)=0\) | | \(\mathcal{V}^{u,\alpha}\) | \(u_*(\alpha)\) | \(u_*(0)=u_*\in(0,\infty)\); \(u_*(\alpha)<\infty\) for all \(\alpha\ge 0\); non-decreasing in \(\alpha\); \(u_*(\alpha)\ge u_*\) | For the occupied phase, positivity of \(\alpha_*(u)\) for every \(u>0\) is proved by comparing \(\mathcal{I}^{u,\alpha}\) with an independent Bernoulli thinning of \(\mathcal{I}^u\) obtained from first-passage holding times of minimal-label trajectories. This yields slab percolation for sufficiently small \(\alpha\): for each fixed \(u>0\), there exist \(R\ge 1\) and \(\alpha_0(u)>0\) such that \(\mathcal{I}^{u,\alpha}\) percolates almost surely in the slab \(\mathbb{Z}^2\times [0,R)^{d-2}\) for all \(\alpha\le \alpha_0(u)\). Finiteness of \(\alpha_*(u)\) follows from Gaussian domination: if \[ \alpha \ge c_5(u):=\frac{(\sqrt{2u}+c_4)^2}{2}, \] then \[ P[B(0,L)\leftrightarrow S(0,2L)\text{ in }\mathcal{I}^{u,\alpha}] \le c e^{-c' L^\rho}. \] For the empty phase, \(u_*(\alpha)\) is finite for every \(\alpha\ge 0\). The argument constructs \((\alpha,u)\)-bad boxes through three local obstructions: a large positive GFF excursion, a large negative GFF excursion, or an unusually small value of \[ L_{y,u}+\frac12\varphi_y^2 \] inside the box. When \(u\) is sufficiently large relative to \(\alpha\), these bad events are rare at all scales, and one obtains \[ P[B(0,L)\leftrightarrow S(0,2L)\text{ in }\mathcal{V}^{u,\alpha}] \le c e^{-c' L^\rho} \] for all \(u\ge c_6(\alpha)\). The thresholds \(\alpha_*(u)\) and \(u_*(\alpha)\) therefore describe a genuine two-parameter phase diagram rather than a trivial deformation of the \(\alpha=0\) model. ## 5. Classical vacant-set percolation and later sharpness results The occupation-time theory sits on top of the classical interlacement geometry. For the unthresholded model, the interlacement set \(\mathcal{I}^u\) is almost surely connected for every \(u>0\), whereas the vacant set \(\mathcal{V}^u\) undergoes a non-trivial phase transition at a threshold \(u_*(d)\in(0,\infty)\) [1410.0605]. The \(\alpha=0\) slice of the occupation-time model is therefore already highly nontrivial on the vacant side, and the thresholded model extends this by allowing one to study under-visited and over-visited regions simultaneously. In the low-intensity regime, local uniqueness for the vacant set was established in all dimensions \(d\ge 3\): for sufficiently small \(u>0\), a large box contains, with stretched-exponentially high probability, a unique macroscopic component of \(\mathcal{V}^u\), and finite vacant clusters have stretched-exponential tail bounds for diameter and volume [1206.6635]. This local picture is exactly the sort of strongly percolative behavior that later became formalized in the threshold \(\bar u\). A major later development is the sharpness theory for the classical vacant set. The phase transition is sharp: for \(u>u_*\), one has stretched-exponential decay of the two-point function, and in \(d\ge 4\) the decay can be made exponential; for \(00\) [2504.02777]. In dimension three the rate depends on \(x\) only through its Euclidean norm at principal exponential order, which offers a precise asymptotic counterpart to the earlier stretched-exponential bounds. ## 6. Extensions, limitations, and open directions Several extensions show that the percolation theory of random interlacements is not confined to the Euclidean lattice. On weighted graphs of the form \(G\times \mathbb{Z}\), under \(\alpha\)-Ahlfors regularity and sub-Gaussian heat-kernel assumptions parameterized by \((\alpha,\beta)\), decoupling inequalities analogous to the lattice case yield \[ 0\le u_* \le u_{**}<\infty, \] as well as stretched-exponential bounds for annular crossing probabilities in the vacant set above \(u_{**}\). In the regime \(\alpha\ge 1+\beta/2\), one further has positivity of the half-plane threshold \(\bar u\), hence \(u_*>0\) [1010.1490]. This suggests that the renormalization philosophy underlying interlacement percolation is robust under substantial geometric inhomogeneity. The model is also stable under small quenched perturbations. If each occupied site of \(\mathcal{I}^u\) is flipped to vacant, and each vacant site to occupied, independently with probability \(\varepsilon\), then for every \(d\ge 3\) and \(u>0\), the perturbed interlacement still percolates almost surely for sufficiently small \(\varepsilon\). On the vacant side, there remains a non-trivial noisy critical threshold \(u_*(\varepsilon)\), and \[ 0<\bar u \le \liminf_{\varepsilon\to 0}u_*(\varepsilon) \le \limsup_{\varepsilon\to 0}u_*(\varepsilon)\le u_{**}<\infty \] for small noise [1109.5086]. A plausible implication is that the large-scale geometry of the interlacement and vacant phases is stable under weak independent disorder, despite the underlying long-range correlations. For the occupation-time level sets of \(\mathcal{I}^{u,\alpha}\) and \(\mathcal{V}^{u,\alpha}\), several issues remain open [1302.7024]. It is not known whether the auxiliary thresholds defined through stretched-exponential crossing decay satisfy \[ \alpha_*(u)=\alpha_{**}(u),\qquad u_*(\alpha)=u_{**}(\alpha). \] The exact values of \(\alpha_*(u)\), \(u_*(\alpha)\), the Gaussian comparison threshold \(c_4\), and the decay exponent \(\rho\) are not determined. For the two-sided Gaussian free field threshold \(\mathfrak h_*\), finiteness is known, but the exact behavior near criticality, uniqueness of the infinite cluster at low levels, and critical exponents remain unresolved. Continuity of the percolation probability in the two-parameter occupation-time model is likewise not addressed there. These unresolved questions define the current frontier of the subject. The established theory shows that occupation-time thresholding turns random interlacements into a genuinely two-parameter correlated percolation model, with exact Gaussian free field comparison, nontrivial occupied and empty phases, and renormalization-controlled subcritical decay. Later sharpness results for the classical vacant set strongly suggest that a comparably sharp description of the full \((u,\alpha)\)-phase diagram should exist, but at present it remains only partially charted. Source: https://www.emergentmind.com/topics/percolation-of-random-interlacements