Abstract
In this paper we study the perturbation theory of model on the whole plane via stochastic quantization. We use integration by parts formula (i.e., Dyson–Schwinger equations) to generate the perturbative expansion for the k-point correlation functions, and prove bounds on the remainder of the truncated expansion using PDE estimates; this in particular proves that the expansion is asymptotic. Furthermore, we derive short distance behaviors of the 2-point function and the connected 4-point function, also via suitable Dyson–Schwinger equations combined with PDE arguments.
Funding Statement
H.S. gratefully acknowledges support by NSF Grants DMS-1954091 and CAREER DMS-2044415.
R.Z. and X.Z. are grateful to the financial supports by National Key R&D Program of China (No. 2022YFA1006300).
R.Z. gratefully acknowledges financial support from the NSFC (No. 12271030).
X.Z. is grateful to the financial supports in part by National Key R&D Program of China (No. 2020YFA0712700) and the NSFC (No. 12090014, 12288201) and the support by key Lab of Random Complex Structures and Data Science, Youth Innovation Promotion Association (2020003), Chinese Academy of Science.
R.Z. and X.Z. are grateful to the financial supports of the financial support by the DFG through the CRC 1283.
Acknowledgments
The authors thank an anonymous referee who brought an important reference [6] to our attention.
Rongchan Zhu is the corresponding author.
Rongchan Zhu is also affiliated with Key Laboratory on MCAACI, Beijing, China.
Citation
Hao Shen. Rongchan Zhu. Xiangchan Zhu. "An SPDE approach to perturbation theory of : Asymptoticity and short distance behavior." Ann. Appl. Probab. 33 (4) 2600 - 2642, August 2023. https://doi.org/10.1214/22-AAP1873
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