Biased 2 x 2 periodic Aztec diamond and an elliptic curve

• 成果类型: Article; Early Access
• 署名作者: Borodin, Alexei; Duits, Maurice
• 署名单位: Massachusetts Institute of Technology (MIT); Royal Institute of Technology
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01195-8
• 发表日期: 2023
• 关键词: STATISTICS; tilings; paths

摘要

We study random domino tilings of the Aztec diamond with a biased 2 x 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.