For $$0<\gamma <2$$ 0 < γ < 2 and $$\delta >0$$ δ > 0 , we consider the Liouville graph distance, which is the minimal number of Euclidean balls of $$\gamma $$ γ -Liouville quantum gravity measure at most $$\delta $$ δ whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at $$\delta \rightarrow 0$$ δ → 0 . In particular, we show that for all $$\delta >0$$ δ > 0 the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints.

中文翻译：

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刘维尔图距离的后续缩放限制

对于 $$0<\gamma <2$$ 0 < γ < 2 和 $$\delta >0$$ δ > 0 ，我们考虑刘维尔图距离，这是 $$\gamma $$ 的最小欧几里得球数γ -Liouville 量子引力测量至多 $$\delta $$ δ，其联合包含两个端点之间的连续路径。在本文中，我们表明重新归一化的距离很紧，因此在 $$\delta \rightarrow 0$$ δ → 0 处具有后续缩放限制。特别是，我们表明，对于所有 $$\delta >0$$ δ > 0，关于 Liouville 图距离的直径与两个端点之间的典型距离具有相同的顺序。