Liouville rst passage percolation(LFPP)with the parameterξ>0 is the family of random distance functions{D_(h)^(ϵ)}ϵ>0 on the plane obtained by integrating e^(ξh),along paths,where{h_(ϵ)}ϵ>0 is a smooth moll...Liouville rst passage percolation(LFPP)with the parameterξ>0 is the family of random distance functions{D_(h)^(ϵ)}ϵ>0 on the plane obtained by integrating e^(ξh),along paths,where{h_(ϵ)}ϵ>0 is a smooth molli cation of the planar Gaussian free eld.Recent works have shown that for allξ>0,the LFPP metrics,appropriately re-scaled,admit non-trivial subsequential limiting metrics.In the caseξ<ξcrit≈0.41,it has been shown that the subsequential limit is unique and de nes a metric onγ-Liouville quantum gravity(LQG)γ=γ(ξ)2(0,2).We prove that for allξ>0,each possible subsequential limiting metric is nearly bi-Lipschitz equivalent to the LFPP metric D_(h)^(ϵ)whenϵis small,even ifϵdoes not belong to the appropriate subsequence.Using this result,we obtain bounds for the scaling constants for LFPP which are sharp up to polylogarithmic factors.We also prove that any two subsequential limiting metrics are bi-Lipschitz equivalent.Our results are an input in subsequent works which shows that the subsequential limits of LFPP induce the same topology as the Euclidean metric whenξ=ξ_(crit)and that the subsequential limit of LFPP is unique whenξ≥ξcrit.展开更多
基金supported by National Science Foundation of USA (Grant Nos. DMS1757479 and DMS-1953848)supported by Clay Research Fellowship
文摘Liouville rst passage percolation(LFPP)with the parameterξ>0 is the family of random distance functions{D_(h)^(ϵ)}ϵ>0 on the plane obtained by integrating e^(ξh),along paths,where{h_(ϵ)}ϵ>0 is a smooth molli cation of the planar Gaussian free eld.Recent works have shown that for allξ>0,the LFPP metrics,appropriately re-scaled,admit non-trivial subsequential limiting metrics.In the caseξ<ξcrit≈0.41,it has been shown that the subsequential limit is unique and de nes a metric onγ-Liouville quantum gravity(LQG)γ=γ(ξ)2(0,2).We prove that for allξ>0,each possible subsequential limiting metric is nearly bi-Lipschitz equivalent to the LFPP metric D_(h)^(ϵ)whenϵis small,even ifϵdoes not belong to the appropriate subsequence.Using this result,we obtain bounds for the scaling constants for LFPP which are sharp up to polylogarithmic factors.We also prove that any two subsequential limiting metrics are bi-Lipschitz equivalent.Our results are an input in subsequent works which shows that the subsequential limits of LFPP induce the same topology as the Euclidean metric whenξ=ξ_(crit)and that the subsequential limit of LFPP is unique whenξ≥ξcrit.
