Exponential trajectories: $M^2$ not matching the theory

[flipdazed]

Issue $M^2$ is not matching the theory. For $m=1$ an exact $\cos^2x$ form is obtained.

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[flipdazed]

More rigorous error analysis

[flipdazed]

The bug was fixed for $m$ in the free field potential and the result is an outwards oscillation for 1M simulations and $m=0.1$

[flipdazed]

Adjusting the separation seems to give more promising results...

[flipdazed]

Running for 1MM simulations

[flipdazed]

Possible issue with the theory used to measure $x^2$?

Is the connected part $$ x^2c = \langle x^2{t,x}\ranglex - \langle x^2{t,x}\rangle $$ or $$ x^2c = x^2{t,x} - \langle x^2{t,x}\rangle $$ with $M^2$ it doesn't matter as $M^2{t,x}$ is scalar anyway

[flipdazed]

Differences in test.sweeper.attemptShort vs. correlations.acorr.acorrMapped

There are two differences between the test case attemptShort and the live case acorrMapped:

1. In acorrMapped I use one array, sep_map for indexing and one for calculation op_samples, whereas in the test I use arr for both

2. I calculate the autocorrelation in the test case as

   ans  += (arr[front] - mean)*(arr[back] - mean)

   but then for some reason I do exactly the same in the live case but also take .mean()

   ans  += ((op_samples[front] - mean)*(op_samples[back] - mean)).mean()

Comments The first difference, 1. does’t look like a problem.

I remember putting some thought into 2. and decided I should be averaging across the lattice sites:

• The magnetisation, $M^2$ is already a scalar quantity so non decaying A/Cs shouldn't be resultant of a .mean() operation as it does nothing to the scalar quantity
• $\phi^2$ just returns the square of all lattice sites and so I calculated $\langle \phi^2\rangle_L$ where $L$ signifies the average over the lattice rather than over HMC time, $t$

Regaining decays in $\phi^2$ If the average $\langle \phi^2 \rangle_L$ is not taken and $\phi^2$ is used: A 2D array of separate autocorrelations corresponding to each lattice site is returned. These individual autocorrelations decay as does their average. It seems that when an average is taken over $L$ in the operator the corresponding A/C looses its decay.

Perhaps there is a phase-space / position space conversion issue in $M^2$ and $\phi^2$ was correct without an averaging?