Conjectured integration-cycle selection on plateaus in the two-dimensional quartic model

Establish that, in the two-dimensional quartic model S(z1,z2) = (λl/4)(z1^2 + z2^2)^2 with l = 5 and kernel Km = e^{-iπ m/24}, the coefficients ai in the integration-cycle decomposition of complex Langevin expectation values satisfy ai = δi,1 on the plateau near m ≈ 10 and ai = δi,2 on the plateau near m ≈ 34, thereby implying that the contributing coefficients in these regions are real integer values.

Background

The paper studies how complex Langevin simulations can converge to incorrect results even when boundary terms vanish, due to contributions from unwanted integration cycles. Building on results that relate complex Langevin expectation values to linear combinations of integrals along independent cycles, the authors investigate the role of a constant kernel parameter Km = e{-iπ m/24} in controlling which cycles are sampled.

After validating the cycle-decomposition picture in a one-dimensional quartic model, the authors extend the analysis to a two-dimensional O(2)-symmetric quartic model S = (λl/4)(z12 + z22)2. They identify two independent integration cycles in this model and observe numerical plateaus in observable behavior near specific kernel parameters. The conjecture asserts that on the first plateau only the real-cycle contribution survives (a1=1, a2=0) and on the second plateau only the imaginary-cycle contribution survives (a2=1, a1=0), implying integer-valued coefficients on these plateaus.

References

We thus conjecture that on the first and second plateau we should find $a_i=\delta_{i,1}$ and $a_i=\delta_{i,2}$, respectively, which also implies that both coefficients are entirely real and integer-valued.

— Kernels and integration cycles in complex Langevin simulations  (2410.13423 - Mandl et al., 2024) in Section 4 (Integration cycles in two dimensions)