Dirac walks on regular trees
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916213374844928 |
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| author | Delporte, Nicolas Sen, Saswato Toriumi, Reiko |
| author_facet | Delporte, Nicolas Sen, Saswato Toriumi, Reiko |
| contents | The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al. [\textit{Laplace and Dirac operators on graphs}, Linear and Multilinear Algebra (2022) 1] proposed a classical "Dirac walk" diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such "Dirac walks" on the Bethe lattice, a $d$-regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ($d_s=1$ for $d=2$ and $d_s=3$ for $d\geq 3$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10881 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dirac walks on regular trees Delporte, Nicolas Sen, Saswato Toriumi, Reiko Statistical Mechanics High Energy Physics - Theory Mathematical Physics The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks. An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al. [\textit{Laplace and Dirac operators on graphs}, Linear and Multilinear Algebra (2022) 1] proposed a classical "Dirac walk" diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such "Dirac walks" on the Bethe lattice, a $d$-regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk ($d_s=1$ for $d=2$ and $d_s=3$ for $d\geq 3$). |
| title | Dirac walks on regular trees |
| topic | Statistical Mechanics High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2312.10881 |