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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].
##### MSC:
 60J65 Brownian motion 60G50 Sums of independent random variables; random walks
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