Our main result is that for any bounded degree graph $X$, the action $W(X)\curvearrowright X$ is diffuse-extensively amenable if and only if $X$ is recurrent. Our proof is based on the construction of suitable stochastic processes $(\tau_t)_{t\geq 0}$ on $W(X)\, <\, \mathfrak{S}(X)$ whose inverted orbits $$\bar O_t(x_0) = \{x\in X\colon \text{there exists } s\leq t \text{\ s.t.\ } \tau_s(x)=x_0\} = \bigcup_{0\leq s \leq t} \tau_s^{-1}(\{x_0\})$$ are exponentially unlikely to be sub-linear when $X$ is transient.

This result leads us to conjecture that the action $W(\mathbb Z^d)\curvearrowright \mathbb Z^d$ is not extensively amenable when $d\geq 3$ and that a different route towards the (non-?)amenability of the IET group may be needed.

我们的主要结果是，对于任何有界度图$ X $，当且仅当$ X $经常出现时，动作$ W（X）\ curvearrowright X $才能广泛地扩散。我们的证明是基于构造合适的随机过程$（\ tau_t）_ {t \ geq 0} $在$ W（X）\，<\，\ mathfrak {S}（X）$上的，其倒转轨道$$ \ bar O_t（x_0）= \ {x \ in X \冒号\ text {存在} s \ leq t \ text {\ st \} \ tau_s（x）= x_0 \} = \ bigcup_ {0 \ leq s \ leq t} \ tau_s ^ {-1}（\ {x_0 \}）$$在$ X $是瞬态时不太可能是次线性的。

此结果使我们推测，当$ d \ geq 3 $时，动作$ W（\ mathbb Z ^ d）\ curvearrowright \ mathbb Z ^ d $不能广泛接受，并且朝向（non-？）可能需要IET组的成员。