Critical exponents for Brownian motion and random walks.
(Exposants critiques pour le mouvement brownien et les marches aléatoires.)

*(French)*Zbl 1012.60072
Séminaire Bourbaki. Volume 1999/2000. Exposés 865-879. Paris: Société Mathématique de France, Astérisque. 276, 29-51, Exp. No. 866 (2002).

The paper has, corresponding to its title, two parts, the first follows Lawler and Werner, the second Kenyon, mainly. Proofs are sketched in Part 1. If \(B_{t}\) is a process, \(B[0,a]\) denotes \(\{B_{t};t\in [0,a]\}\) and \(T_{R}(B)\) the first visit in \((|\cdot |=R)\) of \(B\) (the considered processes have \({\mathbb C}\) as state space). If \(A\subset {\mathbb C}\), consider \(Z_{R}(A)= P(W[0,T_{R}(W)]\cap A=\emptyset)\), where \(W\) is a Brownian motion with \(W_{0}\) uniformly distributed on \((|\cdot |=1)\). For \(\lambda \geq 0\) one defines
\[
\zeta (1,\lambda)= -\lim_{R\rightarrow \infty }\log E(Z_{R}(B[0,T_{R}(B)])^{\lambda }),
\]
where \(B\) is a Brownian motion as \(W\). In an analogous manner one defines “the exponents” \(\zeta (n_{1},\lambda_{1},\dots ,n_{k},\lambda_{k})\), \(n_{i}\geq 1\), \(\lambda_{i}\geq 0\), using a family of \(n_{1}+\dots +n_{k}\) independent Brownians, divided in \(k\) groups, containing \(n_{1},\dots ,n_{k}\) of them respectively and multiplying the \(Z^{\lambda }\)’s. In the same way one defines \(\xi (\lambda_{0},n_{1},\lambda_{1},\dots ,n_{k},\lambda_{k})\) requiring in \(Z_{R}\) also that \(W[0,T_{R}(W)]\subset (\text{Im}>0)\), \(\lambda_{0}\) appearing when \(\subset (\text{Im}>0)\) is the single requirement. The \(\xi\)’s satisfy \(\xi (\lambda_{0},n_{1},\dots ,\lambda_{k})=\xi (\lambda_{0},n_{1},\dots ,n_{j},\xi (\lambda_{j},n_{j+1},\dots ,\lambda_{k}))\), while the \(\zeta\)’s satisfy an analogous relation, with only the first two \(\xi\)’s replaced by \(\zeta\)’s (and without \(\lambda_{0}\)).

Defining also \(\xi (\alpha ,\alpha ')\) by \(\xi (0,1,\xi (\alpha ,\alpha ')) =\xi (\alpha ,1,\alpha ')\), \(\xi (\alpha_{1},\dots ,\alpha_{k})=\xi (\alpha_{1},\xi (\alpha_{2},\dots ,\alpha_{k}))\), \(\zeta (\alpha_{1},\dots ,\alpha_{k}) =\) \(\zeta (1,\xi (\alpha_{1},\dots ,\alpha_{j- 1},\beta_{j},\alpha_{j+1},\dots ,\alpha_{k}))\), \(\alpha_{j}\geq 1\), \(\alpha_{j} =\xi (1,\beta_{j})\), the author states as main theorem the existence of a homeomorphism \(U:[0,\infty)\rightarrow [0,\infty)\) such that \[ \xi (\alpha_{1},\dots ,\alpha_{k})=U^{-1}(U(\alpha_{1})+\dots +U(\alpha_{k})) \] and that \(\zeta (\alpha_{1},\dots ,\alpha_{k})\) is a function of \(\xi (\alpha_{1},\dots ,\alpha_{k})\). As applications of these concepts we mention the study of the Hausdorff dimension of the frontier of the component of \(\infty\) in \(B([0,T_{R}(B)])^{c}\) and of the set of \(z\)’s which disconnect \(B[0,1]\). Then more recent results are mentioned: results allowing explicit calculations of \(\xi\),\(\zeta\),\(U\) and definition of exponents for finite families of measures on the set of all “oriented bridges” between two arcs of \((|\cdot |=1)\) and their relations with the \(\xi\)’s.

The exact title of the second part is: Random walks with deleted loops and domino pavings. The first concept means replacing \((\gamma_n)_{0\leq n\leq N}\) by \((\delta_n)_{0\leq n\leq N'}\) where \(k=\min\{r;\gamma_{r}\in \{\gamma_{0},\dots ,\gamma_{r -1}\}\}\), \(\gamma_{k} =\gamma_{i}\), \(i<k\), \(\delta_{n}=\gamma_{n}\) for \(n\leq i\) and \(\delta_{i+n}=\gamma_{k+n}\) for \(n\geq 1\) and repeating this procedure until possible. For the usual random walks on \({\mathbb N}\times{\mathbb Z}\) and on \({\mathbb Z}^2\) it was shown to be possible to define a corresponding random walk with deleted loops (RWDL) and the probability that \(r e^{i\theta }\) is visited by that on \({\mathbb N}\times{\mathbb Z}\) starting from \(0\) is \(r^{-(3/4)(1+o(1))}((\cos\theta)^{1/4}+o(1))\) for \(r\rightarrow \infty\). It is shown that the RWDL starting from \(0\), on \({\mathbb Z}^{d}\), \(d\leq 4\), and on \({\mathbb N}\times {\mathbb Z}\), has as law the limit of the uniform distribution on the set of all trees which are subgraphs of \([-n,n]^{d}\cap {\mathbb Z}^{d}\), \([-n,n ]^{2}\cap ({\mathbb N}\times {\mathbb Z})\) containing all its vertices.

Consider now the “chess paving” of \({\mathbb R}^2\) composed of unit squares, the set of their centers being \({\mathbb Z}^2\) and such that the one centered in \((0,0)\) is white. Let \(W_{0}\) be the set of all white squares with centers \((2m,2n)\), \(W_{1}\) the set of all other white ones, \(B_{0}\) the set of all blacks with centers \((2m+1,2n+1)\), \(B_{1}\) the set of all other blacks. A temperlien polyomino is a “union \(P\) of squares of this paving, having as frontier a simple curve” (property (u)), such that all corners correspond to squares in \(B_{1}\) and from which a black square on the frontier is deleted (named the basic square of \(P\)). If \(M(P)\) is the set of its squares, one defines the matrix \(K (v,w)\), \(v,w\in M\), by \(K(v,w)=1,i,-1,-i\) if \(w\) is the right, upper, left, lower neighbor of \(v\), respectively, and \(0\) otherwise. Let \(C(P)= K^{-1}\). Then \(|C(P)(v,w)|\) is the probability that, choosing at random a decomposition of \(P\) into “dominoes” (i.e. unions of two adjacent unit squares), the domino \(vw\) figures in it. Let \(U\) have property (u) and be approximated (\(\varepsilon \rightarrow 0\)) by \(\varepsilon P_{\varepsilon }\) where \(P_{\varepsilon }\) are temperlien polyominoes (corner by corner and with basic points convergent to a \(d_{0}\)) and let \(\varepsilon v_{\varepsilon }\rightarrow v\), \(\varepsilon w_{\varepsilon }\rightarrow w\), \(v,w\in U\), \(v_{\varepsilon },w_{\varepsilon }\in M(P_{\varepsilon })\). Then \(\lim_{\varepsilon \rightarrow 0}C(P_{\varepsilon }) (v_{\varepsilon },w_{\varepsilon })\) exists and equals \(\text{Re} F_{0}(v,w)\), \(i\text{Im} F_{0}(v,w)\), \(\text{Re}F_{1}(v,w)\), \(i\text{Im} F_{1}(v,w)\) if, for all \(\varepsilon\), “\(v_{\varepsilon }\in W_{0}\), \(w_{\varepsilon }\in B_{0}\)”, “\(v_{\varepsilon }\in W_{0}\), \(w_{\varepsilon }\in B_{1}\)”, “\(v_{\varepsilon }\in W_{1}\), \(w_{\varepsilon }\in B_{0}\)”, “\(v_{\varepsilon }\in W_{1}\), \(w_{\varepsilon }\in B_{1}\)”, respectively; \(F_{0}(v,\cdot)\) and \(F_{1}(v,\cdot)\) are meromorphic, null in \(d_{0}\) having each a single pole in \(v\) with residue \(1/\pi\), \(\text{Re}F_{0}(v,\cdot)\) and \(\text{Im}F_{1}(v,\cdot)\) being null on \(\partial U\). The final theorem is an asymptotic expression for the number of decompositions into dominoes of \(P_{\varepsilon}\), containing \(o(\log(1/\varepsilon))\) when \(U\) is not a rectangle and \(o (1)\) when it is. The paper contains other comments and relations with physics, particularly with quantum gravitation.

For the entire collection see [Zbl 0981.00011].

Defining also \(\xi (\alpha ,\alpha ')\) by \(\xi (0,1,\xi (\alpha ,\alpha ')) =\xi (\alpha ,1,\alpha ')\), \(\xi (\alpha_{1},\dots ,\alpha_{k})=\xi (\alpha_{1},\xi (\alpha_{2},\dots ,\alpha_{k}))\), \(\zeta (\alpha_{1},\dots ,\alpha_{k}) =\) \(\zeta (1,\xi (\alpha_{1},\dots ,\alpha_{j- 1},\beta_{j},\alpha_{j+1},\dots ,\alpha_{k}))\), \(\alpha_{j}\geq 1\), \(\alpha_{j} =\xi (1,\beta_{j})\), the author states as main theorem the existence of a homeomorphism \(U:[0,\infty)\rightarrow [0,\infty)\) such that \[ \xi (\alpha_{1},\dots ,\alpha_{k})=U^{-1}(U(\alpha_{1})+\dots +U(\alpha_{k})) \] and that \(\zeta (\alpha_{1},\dots ,\alpha_{k})\) is a function of \(\xi (\alpha_{1},\dots ,\alpha_{k})\). As applications of these concepts we mention the study of the Hausdorff dimension of the frontier of the component of \(\infty\) in \(B([0,T_{R}(B)])^{c}\) and of the set of \(z\)’s which disconnect \(B[0,1]\). Then more recent results are mentioned: results allowing explicit calculations of \(\xi\),\(\zeta\),\(U\) and definition of exponents for finite families of measures on the set of all “oriented bridges” between two arcs of \((|\cdot |=1)\) and their relations with the \(\xi\)’s.

The exact title of the second part is: Random walks with deleted loops and domino pavings. The first concept means replacing \((\gamma_n)_{0\leq n\leq N}\) by \((\delta_n)_{0\leq n\leq N'}\) where \(k=\min\{r;\gamma_{r}\in \{\gamma_{0},\dots ,\gamma_{r -1}\}\}\), \(\gamma_{k} =\gamma_{i}\), \(i<k\), \(\delta_{n}=\gamma_{n}\) for \(n\leq i\) and \(\delta_{i+n}=\gamma_{k+n}\) for \(n\geq 1\) and repeating this procedure until possible. For the usual random walks on \({\mathbb N}\times{\mathbb Z}\) and on \({\mathbb Z}^2\) it was shown to be possible to define a corresponding random walk with deleted loops (RWDL) and the probability that \(r e^{i\theta }\) is visited by that on \({\mathbb N}\times{\mathbb Z}\) starting from \(0\) is \(r^{-(3/4)(1+o(1))}((\cos\theta)^{1/4}+o(1))\) for \(r\rightarrow \infty\). It is shown that the RWDL starting from \(0\), on \({\mathbb Z}^{d}\), \(d\leq 4\), and on \({\mathbb N}\times {\mathbb Z}\), has as law the limit of the uniform distribution on the set of all trees which are subgraphs of \([-n,n]^{d}\cap {\mathbb Z}^{d}\), \([-n,n ]^{2}\cap ({\mathbb N}\times {\mathbb Z})\) containing all its vertices.

Consider now the “chess paving” of \({\mathbb R}^2\) composed of unit squares, the set of their centers being \({\mathbb Z}^2\) and such that the one centered in \((0,0)\) is white. Let \(W_{0}\) be the set of all white squares with centers \((2m,2n)\), \(W_{1}\) the set of all other white ones, \(B_{0}\) the set of all blacks with centers \((2m+1,2n+1)\), \(B_{1}\) the set of all other blacks. A temperlien polyomino is a “union \(P\) of squares of this paving, having as frontier a simple curve” (property (u)), such that all corners correspond to squares in \(B_{1}\) and from which a black square on the frontier is deleted (named the basic square of \(P\)). If \(M(P)\) is the set of its squares, one defines the matrix \(K (v,w)\), \(v,w\in M\), by \(K(v,w)=1,i,-1,-i\) if \(w\) is the right, upper, left, lower neighbor of \(v\), respectively, and \(0\) otherwise. Let \(C(P)= K^{-1}\). Then \(|C(P)(v,w)|\) is the probability that, choosing at random a decomposition of \(P\) into “dominoes” (i.e. unions of two adjacent unit squares), the domino \(vw\) figures in it. Let \(U\) have property (u) and be approximated (\(\varepsilon \rightarrow 0\)) by \(\varepsilon P_{\varepsilon }\) where \(P_{\varepsilon }\) are temperlien polyominoes (corner by corner and with basic points convergent to a \(d_{0}\)) and let \(\varepsilon v_{\varepsilon }\rightarrow v\), \(\varepsilon w_{\varepsilon }\rightarrow w\), \(v,w\in U\), \(v_{\varepsilon },w_{\varepsilon }\in M(P_{\varepsilon })\). Then \(\lim_{\varepsilon \rightarrow 0}C(P_{\varepsilon }) (v_{\varepsilon },w_{\varepsilon })\) exists and equals \(\text{Re} F_{0}(v,w)\), \(i\text{Im} F_{0}(v,w)\), \(\text{Re}F_{1}(v,w)\), \(i\text{Im} F_{1}(v,w)\) if, for all \(\varepsilon\), “\(v_{\varepsilon }\in W_{0}\), \(w_{\varepsilon }\in B_{0}\)”, “\(v_{\varepsilon }\in W_{0}\), \(w_{\varepsilon }\in B_{1}\)”, “\(v_{\varepsilon }\in W_{1}\), \(w_{\varepsilon }\in B_{0}\)”, “\(v_{\varepsilon }\in W_{1}\), \(w_{\varepsilon }\in B_{1}\)”, respectively; \(F_{0}(v,\cdot)\) and \(F_{1}(v,\cdot)\) are meromorphic, null in \(d_{0}\) having each a single pole in \(v\) with residue \(1/\pi\), \(\text{Re}F_{0}(v,\cdot)\) and \(\text{Im}F_{1}(v,\cdot)\) being null on \(\partial U\). The final theorem is an asymptotic expression for the number of decompositions into dominoes of \(P_{\varepsilon}\), containing \(o(\log(1/\varepsilon))\) when \(U\) is not a rectangle and \(o (1)\) when it is. The paper contains other comments and relations with physics, particularly with quantum gravitation.

For the entire collection see [Zbl 0981.00011].

Reviewer: Ion Cuculescu (Bucureşti)