Abstract
We prove that the rescaled one-point fluctuations of the boundary of the percolation cluster in the Bernoulli-Exponential first passage percolation around the diagonal converge to a new family of distributions. The limit law is indexed by the rescaled level of percolation , it is Gaussian for and it converges to the Tracy–Widom distribution as . For a fixed level the width of the cluster in the limit as a function of a time parameter t is of order with Tracy–Widom fluctuations as in the discrete model.
Acknowledgments
We thank Bálint Virág and Patrik Ferrari for discussions about polymer models and correlation kernels and Guillaume Barraquand for pointing out the scaling in Theorem 1.3 and for his comments. The work of the author was supported by the NKFI (National Research, Development and Innovation Office) grants FK142124 and KKP144059 “Fractal geometry and applications”, by the Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the ÚNKP–22–5–BME–250 New National Excellence Program of the Ministry for Innovation and Technology from the source of the NKFI.
Citation
Bálint Vető. "Fluctuations around the diagonal in Bernoulli-Exponential first passage percolation." Electron. Commun. Probab. 29 1 - 14, 2024. https://doi.org/10.1214/24-ECP601
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