## Charts: Energy vs. Number of Points & Ratio of Scores
### Overview
The image presents two charts side-by-side. The left chart displays "Energy" as a function of "Number of Points N" for two different methods: "Rakhmanov-Saff-Zhou Asymptotics" and "AlphaEvolve". The right chart shows the "Ratio" of "AlphaEvolve-score" to "Rakhmanov-Saff-Zhou Asymptotics ratio" also as a function of "Number of Points N". Both charts share the same x-axis.
### Components/Axes
* **Left Chart:**
* X-axis: "Number of Points N" (Scale: approximately 100 to 1000, with markers at 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000)
* Y-axis: "Energy" (Scale: approximately 0 to 50000, with markers at 0, 10000, 20000, 30000, 40000, 50000)
* Data Series:
* "Rakhmanov-Saff-Zhou Asymptotics" (Blue line)
* "AlphaEvolve" (Orange dashed line)
* **Right Chart:**
* X-axis: "Number of Points N" (Scale: approximately 100 to 1000, with markers at 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000)
* Y-axis: "Ratio" (Scale: approximately 0.99992 to 1.00008, with markers at 0.99992, 0.99996, 1.00000, 1.00004, 1.00008)
* Data Series:
* "AlphaEvolve-score / Rakhmanov-Saff-Zhou Asymptotics ratio" (Light blue line)
### Detailed Analysis or Content Details
* **Left Chart:**
* Rakhmanov-Saff-Zhou Asymptotics (Blue line): The line slopes upward consistently.
* At N = 100, Energy ≈ 1000
* At N = 200, Energy ≈ 4000
* At N = 300, Energy ≈ 9000
* At N = 400, Energy ≈ 16000
* At N = 500, Energy ≈ 25000
* At N = 600, Energy ≈ 34000
* At N = 700, Energy ≈ 42000
* At N = 800, Energy ≈ 48000
* At N = 900, Energy ≈ 49000
* At N = 1000, Energy ≈ 50000
* AlphaEvolve (Orange dashed line): The line also slopes upward consistently, but is generally below the blue line until N = 900, where it surpasses it.
* At N = 100, Energy ≈ 500
* At N = 200, Energy ≈ 2000
* At N = 300, Energy ≈ 5000
* At N = 400, Energy ≈ 10000
* At N = 500, Energy ≈ 16000
* At N = 600, Energy ≈ 23000
* At N = 700, Energy ≈ 31000
* At N = 800, Energy ≈ 38000
* At N = 900, Energy ≈ 46000
* At N = 1000, Energy ≈ 49000
* **Right Chart:**
* AlphaEvolve-score / Rakhmanov-Saff-Zhou Asymptotics ratio (Light blue line): The line fluctuates around 1, with dips and peaks.
* At N = 100, Ratio ≈ 1.00006
* At N = 200, Ratio ≈ 1.00002
* At N = 300, Ratio ≈ 0.99998
* At N = 400, Ratio ≈ 0.99994
* At N = 500, Ratio ≈ 0.99996
* At N = 600, Ratio ≈ 0.99992
* At N = 700, Ratio ≈ 0.99994
* At N = 800, Ratio ≈ 0.99990
* At N = 900, Ratio ≈ 0.99994
* At N = 1000, Ratio ≈ 0.99992
### Key Observations
* The energy increases with the number of points for both methods.
* AlphaEvolve has lower energy values than Rakhmanov-Saff-Zhou Asymptotics for most values of N, but surpasses it at N = 1000.
* The ratio of AlphaEvolve score to Rakhmanov-Saff-Zhou Asymptotics ratio oscillates around 1, indicating that the two methods produce similar results, but with some variation. The ratio dips below 1 for some values of N, suggesting that AlphaEvolve's score is slightly lower than the asymptotic prediction.
### Interpretation
The charts compare the performance of two methods, "Rakhmanov-Saff-Zhou Asymptotics" and "AlphaEvolve", in terms of energy calculation as the number of points increases. The left chart shows that both methods yield increasing energy values with more points, as expected. The right chart provides a normalized comparison, showing how closely AlphaEvolve's results align with the asymptotic prediction. The fluctuations around 1 suggest that AlphaEvolve is a good approximation of the asymptotic behavior, but not perfectly aligned. The fact that AlphaEvolve surpasses the asymptotic prediction at higher N (N=1000) could indicate that AlphaEvolve is more accurate for larger datasets, or that the asymptotic formula is less accurate in that regime. The consistent oscillation of the ratio suggests a systematic difference between the two methods, rather than random noise. This could be due to the specific implementation of AlphaEvolve or inherent limitations of the asymptotic approximation.
