$\mathrm{U}(N)$ lattice Yang-Mills in the 't Hooft regime
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914116189290496 |
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| author | Nissim, Ron |
| author_facet | Nissim, Ron |
| contents | We establish a mass gap, prove the existence of a unique infinite volume limit, and give a new proof of the large $N$ limit for $\mathrm{U}(N)$ lattice Yang-Mills theory in the 't Hooft regime. These results were previously obtained for $\mathrm{SU}(N)$ and $\mathrm{SO}(N)$ lattice Yang-Mills theories as applications of the mixing of the associated Langevin dynamics, which is verified via the Bakry-Émery criterion [SZZ23]. For $\mathrm{U}(N)$, however, this approach fails because its Ricci curvature is not uniformly positive, and as a result the Bakry-Émery condition cannot be easily verified. To overcome this obstacle, we recast the $\mathrm{U}(N)$ theory as a random-environment $\mathrm{SU}(N)$ model, where the randomness arises from a $\mathrm{U}(1)$ field, and combine cluster-expansion and Langevin-dynamics techniques to analyze the resulting $\mathrm{U}(1)\times\mathrm{SU}(N)$ model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_22788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathrm{U}(N)$ lattice Yang-Mills in the 't Hooft regime Nissim, Ron Probability Mathematical Physics We establish a mass gap, prove the existence of a unique infinite volume limit, and give a new proof of the large $N$ limit for $\mathrm{U}(N)$ lattice Yang-Mills theory in the 't Hooft regime. These results were previously obtained for $\mathrm{SU}(N)$ and $\mathrm{SO}(N)$ lattice Yang-Mills theories as applications of the mixing of the associated Langevin dynamics, which is verified via the Bakry-Émery criterion [SZZ23]. For $\mathrm{U}(N)$, however, this approach fails because its Ricci curvature is not uniformly positive, and as a result the Bakry-Émery condition cannot be easily verified. To overcome this obstacle, we recast the $\mathrm{U}(N)$ theory as a random-environment $\mathrm{SU}(N)$ model, where the randomness arises from a $\mathrm{U}(1)$ field, and combine cluster-expansion and Langevin-dynamics techniques to analyze the resulting $\mathrm{U}(1)\times\mathrm{SU}(N)$ model. |
| title | $\mathrm{U}(N)$ lattice Yang-Mills in the 't Hooft regime |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2510.22788 |