Feynman Checkers: an Asymptotic Formula for the Wave Function

Student: Kuyanov Fedor

Educational Programme: Applied Mathematics and Information Science (Bachelor)

Final Grade: 10

Year of Graduation: 2024

This work aims to investigate the asymptotic behavior of the wave function in Feynman checkers, which is an elementary mathematical model of electron motion along a line. In this model, our world is viewed as an infinite checkerboard, where a checker moves from one point to another, counting each turn. This model is also known as a one-dimensional quantum walk, an Ising model at imaginary temperature, or a special case of the six-vertex model with complex coefficients. We prove a new asymptotic formula for the wave function in terms of the Airy function, which is uniform for $|x / t| < 1 - \delta$. The used techniques involve Fourier transform, complex analysis, and conformal mappings.

Full text (added May 20, 2024)

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