## Influence and sharp-threshold theorems for monotonic measures.(English)Zbl 1115.60099

Let $$\Omega=\{0,1\}^n$$ or $$[0,1]^n$$, let $$K=P(\Omega)$$ or the Lebesgue $$\sigma$$-algebra of $$\Omega$$, and let $$\mu$$ be a probability on $$K$$ such that $$\mu(\{\omega\})>0$$, $$\omega\in\Omega$$, if $$\Omega=\{0,1\}^n$$, while $$\mu(A)=\int_{A}\rho \,d\lambda$$, $$A\in K$$, if $$\Omega=[0,1]^n$$, where $$\lambda$$ is the Lebesgue measure on $$K$$ and $$\rho$$ is a positive Lebesgue measurable density. $$\mu$$ is assumed to satisfy a certain monotonicity property. Put $$I=\{1,\dots,n\}$$. For $$\omega=(\omega_1,\dots,\omega_n)\in\Omega$$ and $$i\in I$$, define $$X_{i}(\omega)=\omega_{i}$$. A set $$A\in K$$ is called increasing if $$\omega\in A$$ whenever $$\omega'\in A$$ and $$\omega\geq\omega'$$. If $$A$$ is increasing and $$i\in I$$, the quantities $$I_{A}(i)=\mu(A\mid X_{i}=1)-\mu(A\mid X_{i}=0)$$ are called influences. Here are the main results.
Theorem 1. There exists an absolute constant $$c\in(0,\infty)$$ such that, for all $$n$$ and all increasing $$A$$, there is $$i\in I$$ satisfying $$I_{A}(i)\geq c(\mu(A)\wedge(1-\mu(A))(\log n)/n$$.
Theorem 2. Let $$\Pi$$ be a subgroup of the group of all permutations of $$I$$ such that for any $$i,j\in I$$ there is $$\pi\in\Pi$$ with $$\pi_{i}=j$$, and that $$\mu(\{\omega\})=\mu(\{\pi(\omega)\})$$ for all $$\omega\in\Omega$$ and $$\pi\in\Pi$$, where $$\pi(\omega_1,\dots,\omega_n)=(\omega_{\pi_1},\dots,\omega_{\pi_{n}})$$, $$\omega\in\Omega$$. For $$\Omega=\{0,1\}^n$$ and $$p\in(0,1)$$, define the probability $$\mu_{p}$$ on $$K$$ by $$\mu_{p}(\{\omega\})=C\prod_{i=1}^n p^{\omega_i}(1-p)^{1-\omega_i}$$, $$\omega=(\omega_1,\dots,\omega_n)\in\Omega$$, where $$C$$ is the normalizing constant. Then there exists an absolute constant $$c\in(0,\infty)$$ such that, for all $$n$$ and all increasing $$A$$ with $$A=\pi(A)$$, $$\pi\in\Pi$$, one has $$(d/(dp))\mu_{p}(A)\geq c\xi_{p}(\mu_{p}(A)\wedge(1-\mu_{p}(A))\log n$$, where $$\xi_{p}=\int_\Omega X_I \,d\mu_{p}(1-\int_\Omega X_I \,d\mu_{p})/p(1-p)$$. These results have applications to problems of discrete probability such as the random cluster model.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60E15 Inequalities; stochastic orderings 82B31 Stochastic methods applied to problems in equilibrium statistical mechanics 82B43 Percolation
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