## TracyWidomBeta.jl

This package aims to compute the Tracy-Widom distribution for arbitrary positive beta.
Author Yiting687691
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February 2022

# TracyWidomBeta

Compute both the cdf and pdf of the Tracy-Widom distribution for arbitrary β>0.

## Example Usage

```F_cdf=TW(2)
F_cdf(0.5)
F_pdf=TW(2;pdf=true)
F_pdf(0.5)
plot(F_cdf)
plot(F_pdf)```

The above code uses the default finite-difference discretization with trapezoidal method. It returns either cdf or pdf, if you set the argument 'pdf' to be true, of the Tracy-Widom distribution function for β=2, and then evaluates the function at x=0.5. Both cdf and pdf can be plotted directly for x between -10 and 13. The pdf can also be obtained by the following way.

`F_pdf=F_cdf'`

If higher accuracy is preferred, specify the arguments as the following, where I choose to apply BDF5 on spectral discretization.

`F_cdf=TW(2;method="spectral",step="bdf5")`

Note that for β=4, one needs to account for an extra 2^(1/6) factor when evaluating at the desired value of x. For example, if one wants to compute the cdf of the Tracy-Widom distribution for β=4 at x=-2 using the finite-difference discretization with trapezoidal method, use the following code.

```F_cdf=TW(4)
F_cdf(-2/(2^(1/6)))```

Also note that the default finite-difference discretization only works well for 1⩽β⩽400, and the more accurate spectral discretization works well for 1⩽β⩽10. For larger values of β, one needs to use finer grid and more Chebyshev points (both discretizations use 1000 Chebyshev points by default). For β<1, one needs to enlarge the domain for x.

By default, all of the above examples use Fourier interpolation to generate a continuous function. One can also get only the discrete values before interpolation. For example, the following code shows how to get the discrete values of the cdf using the finite-difference discretization with trapezoidal method and the corresponding values of x.

```F_cdf=TW(2;interp=false)
x=F[1]
F_cdf=F[2]```

See Index for a brief summary of how this algorithm is created and related plots.

## References

T. Trogdon & Y. Zhang (2023), Computing the Tracy-Widom distribution for arbitrary β>0, arXiv:2304.04951

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