Scaling limit and tail bounds for a random walk model of SOS level lines
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916614413221888 |
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| author | Hegde, Milind Kim, Yujin H. Serio, Christian |
| author_facet | Hegde, Milind Kim, Yujin H. Serio, Christian |
| contents | This paper analyzes a random walk model for the level lines appearing in the entropic repulsion phenomena of three-dimensional discrete random interfaces above a hard wall; we are particularly motivated by the low-temperature (2+1)D solid-on-solid (SOS) model, where the emergence of these level lines has been rigorously established. The model we consider is a line ensemble of non-crossing random walk bridges above a wall with geometrically growing area tilts. Our main result, which in particular resolves a question of Caputo, Ioffe, and Wachtel (2019), is an edge 1:2:3 scaling limit for this ensemble as the domain size $N$ diverges, with a growing number of walks (including the number of level lines of the SOS model) and high boundary conditions (covering the maximum upper deviation of the SOS level lines). As a key input, we establish Tracy--Widom-type upper tail bounds for each of the relevant curves in the line ensemble. An ingredient which may be of independent interest is a ballot theorem for random walk bridges under a broader range of boundary values than available in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10384 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scaling limit and tail bounds for a random walk model of SOS level lines Hegde, Milind Kim, Yujin H. Serio, Christian Probability Mathematical Physics 60G50, 60F17, 82B41, 82B20 This paper analyzes a random walk model for the level lines appearing in the entropic repulsion phenomena of three-dimensional discrete random interfaces above a hard wall; we are particularly motivated by the low-temperature (2+1)D solid-on-solid (SOS) model, where the emergence of these level lines has been rigorously established. The model we consider is a line ensemble of non-crossing random walk bridges above a wall with geometrically growing area tilts. Our main result, which in particular resolves a question of Caputo, Ioffe, and Wachtel (2019), is an edge 1:2:3 scaling limit for this ensemble as the domain size $N$ diverges, with a growing number of walks (including the number of level lines of the SOS model) and high boundary conditions (covering the maximum upper deviation of the SOS level lines). As a key input, we establish Tracy--Widom-type upper tail bounds for each of the relevant curves in the line ensemble. An ingredient which may be of independent interest is a ballot theorem for random walk bridges under a broader range of boundary values than available in the literature. |
| title | Scaling limit and tail bounds for a random walk model of SOS level lines |
| topic | Probability Mathematical Physics 60G50, 60F17, 82B41, 82B20 |
| url | https://arxiv.org/abs/2502.10384 |