Uniqueness of the critical and supercritical Liouville quantum gravity metrics
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arXiv
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| Format: | Preprint |
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2021
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| author | Ding, Jian Gwynne, Ewain |
| author_facet | Ding, Jian Gwynne, Ewain |
| contents | We show that for each ${\mathbf c}_{\mathrm M} \in [1,25)$, there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge ${\mathbf c}_{\mathrm M}$. An earlier series of works by Ding-Dubédat-Dunlap-Falconet, Gwynne-Miller, and others showed that such a metric exists and is unique in the subcritical case ${\mathbf c}_{\mathrm M} \in (-\infty,1)$, which corresponds to coupling constant $γ\in (0,2)$. The critical case ${\mathbf c}_{\mathrm M} = 1$ corresponds to $γ=2$ and the supercritical case ${\mathbf c}_{\mathrm M} \in (1,25)$ corresponds to $γ\in \mathbb C$ with $|γ| = 2$.
Our metric is constructed as the limit of an approximation procedure called Liouville first passage percolation, which was previously shown to be tight for $\mathbf c_{\mathrm M} \in [1,25)$ by Ding and Gwynne (2020). In this paper, we show that the subsequential limit is uniquely characterized by a natural list of axioms. This extends the characterization of the LQG metric proven by Gwynne and Miller (2019) for $\mathbf c_{\mathrm M} \in (-\infty,1)$ to the full parameter range $\mathbf c_{\mathrm M} \in (-\infty,25)$.
Our argument is substantially different from the proof of the characterization of the LQG metric for $\mathbf c_{\mathrm M} \in (-\infty,1)$. In particular, the core part of the argument is simpler and does not use confluence of geodesics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_00177 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Uniqueness of the critical and supercritical Liouville quantum gravity metrics Ding, Jian Gwynne, Ewain Probability Mathematical Physics Metric Geometry We show that for each ${\mathbf c}_{\mathrm M} \in [1,25)$, there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge ${\mathbf c}_{\mathrm M}$. An earlier series of works by Ding-Dubédat-Dunlap-Falconet, Gwynne-Miller, and others showed that such a metric exists and is unique in the subcritical case ${\mathbf c}_{\mathrm M} \in (-\infty,1)$, which corresponds to coupling constant $γ\in (0,2)$. The critical case ${\mathbf c}_{\mathrm M} = 1$ corresponds to $γ=2$ and the supercritical case ${\mathbf c}_{\mathrm M} \in (1,25)$ corresponds to $γ\in \mathbb C$ with $|γ| = 2$. Our metric is constructed as the limit of an approximation procedure called Liouville first passage percolation, which was previously shown to be tight for $\mathbf c_{\mathrm M} \in [1,25)$ by Ding and Gwynne (2020). In this paper, we show that the subsequential limit is uniquely characterized by a natural list of axioms. This extends the characterization of the LQG metric proven by Gwynne and Miller (2019) for $\mathbf c_{\mathrm M} \in (-\infty,1)$ to the full parameter range $\mathbf c_{\mathrm M} \in (-\infty,25)$. Our argument is substantially different from the proof of the characterization of the LQG metric for $\mathbf c_{\mathrm M} \in (-\infty,1)$. In particular, the core part of the argument is simpler and does not use confluence of geodesics. |
| title | Uniqueness of the critical and supercritical Liouville quantum gravity metrics |
| topic | Probability Mathematical Physics Metric Geometry |
| url | https://arxiv.org/abs/2110.00177 |