Deborah.Miriam.Cumulants

calc_Q(
    trMi::Vector{T}
) where T -> (real(T), real(T), real(T), real(T))

Compute the first four moments $Q_n \; (n=1,2,3,4)$ from $\text{Tr} \, M^{-n} \; (n=1,2,3,4)$ vectors.

Arguments

• trMi::Vector{T}: Vector containing traces, where
  • trMi[2] is $\text{Tr} \, M^{-1}$
  • trMi[3] is $\text{Tr} \, M^{-2}$
  • trMi[4] is $\text{Tr} \, M^{-3}$
  • trMi[5] is $\text{Tr} \, M^{-4}$

Formula

\[\begin{aligned} Q_1 &= N_{\text{f}} \, \text{Tr} \, M^{-1} \\ Q_2 &= - N_{\text{f}} \, \text{Tr} \, M^{-2} + \left( \text{Tr} \, M^{-1} \right)^2 \\ Q_3 &= 2 \, N_{\text{f}} \, \text{Tr} \, M^{-3} - 3 \, N_{\text{f}}^{2} \, \text{Tr} \, M^{-2} \, \text{Tr}\, M^{-1} + \left( N_{\text{f}} \, \text{Tr} \, M^{-1} \right)^{3} \\ Q_4 &= - 6 \, N_{\text{f}} \, \text{Tr} \, M^{-4} + 8 \, N_{\text{f}}^{2} \, \text{Tr} \, M^{-3} \, \text{Tr} \, M^{-1} + 3 \left( N_{\text{f}} \, \text{Tr} \, M^{-2} \right)^{2} - 6 \, N_{\text{f}} \, \text{Tr} \, M^{-2} \, \left( N_{\text{f}} \, \text{Tr} \, M^{-1} \right)^{2} + \left( N_{\text{f}} \, \text{Tr} \, M^{-1} \right)^{4} \end{aligned}\]

Notes

Unlike this Deborah.Miriam, Deborah.Esther uses slightly different definitions for $Q_{n} \; (n=1,2,3,4)$. In this implementation, the factor of $N_{\text{f}}$ does not explicitly appear, because the trace rescaling step already incorporates $N_{\text{f}}$ into the normalization.

See also:

• Deborah.Miriam.MultiEnsembleLoader.generate_trMi_vector
• Deborah.Esther.TraceRescaler.rescale_trace (for comparison of rescaling conventions)

Returns

• Q1::T: $Q_1$ moment
• Q2::T: $Q_2$ moment
• Q3::T: $Q_3$ moment
• Q4::T: $Q_4$ moment

@inline @fastmath calc_cumulants(
    V::T, 
    Q1::T, 
    Q2::T, 
    Q3::T, 
    Q4::T, 
    jobid::Union{Nothing, String}=nothing
) where T -> (T, T, T, T)

Compute statistical cumulants from trace moments $Q_1$ through $Q_4$.

Arguments

• V::T: Volume (typically $N_S^3 \times N_T$), used for normalization.
• Q1::T: First moment $Q_1$
• Q2::T: Second moment $Q_2$
• Q3::T: Third moment $Q_3$
• Q4::T: Fourth moment $Q_4$
• jobid::Union{Nothing, String} : Optional job ID for logging.

Formula

\[\begin{align*} \chi &= \frac{\left\langle Q_2 \right\rangle - \left\langle Q_1 \right\rangle^2}{V} \\ S &= \frac{ \left\langle Q_3 \right\rangle - 3 \, \left\langle Q_2 \right\rangle \, \left\langle Q_1 \right\rangle + 2 \, \left\langle Q_1 \right\rangle^3 }{\left( \left\langle Q_2 \right\rangle - \left\langle Q_1 \right\rangle^2 \right)^{\frac{3}{2}}} \\ K &= \frac{ \left\langle Q_4 \right\rangle - 4 \, \left\langle Q_3 \right\rangle \, \left\langle Q_1 \right\rangle - 3 \, \left\langle Q_2 \right\rangle^2 + 12 \, \left\langle Q_2 \right\rangle \, \left\langle Q_1 \right\rangle^2 - 6 \, \left\langle Q_1 \right\rangle^4 }{\left( \left\langle Q_2 \right\rangle - \left\langle Q_1 \right\rangle^2 \right)^{2}} \\ B &= 1 - \frac{\left\langle Q_4 \right\rangle}{3 \, \left\langle Q_2 \right\rangle^2} \end{align*}\]

Returns

• susceptibility::T: susceptibility of chiral condensate ($\chi$)
• skewness::T: skewness of chiral condensate (measures asymmetry)
• kurtosis::T: kurtosis of chiral condensate (measures tailedness)
• binder_cumulant::T: Binder cumulant

Notes

• Prevents division by near-zero variance using eps(T) safeguard.
• Assumes input types are floating point (e.g., Float64, Float32).

calc_trace(
    trMi::Vector{T}
) where T -> (real(T), real(T), real(T), real(T))

Extract the first four raw trace components ($\text{Tr} \, M^{-n} \; (n=1,2,3,4)$) from a trace vector.

Arguments

• trMi::Vector{T}: Trace vector where indices 2..5 contain the four trace components:
  • trMi[2] is $\text{Tr} \, M^{-1}$
  • trMi[3] is $\text{Tr} \, M^{-2}$
  • trMi[4] is $\text{Tr} \, M^{-3}$
  • trMi[5] is $\text{Tr} \, M^{-4}$

Returns

• trM1::real(T), trM2::real(T), trM3::real(T), trM4::real(T)

Notes

• Requires length(trMi) $\ge 5$.

compute_cumulants(
    bin_size::Int, 
    V::Int, 
    obs::Vector{Vector{T}}, 
    w::Vector{T}, 
    jobid::Union{Nothing, String}=nothing
) where T -> Vector{Vector{T}}

Compute raw jackknife-resampled cumulants from reweighted trace observables.

This function performs jackknife resampling of reweighted $Q$-moments ($Q_{n} \; (n=1,2,3,4)$), and computes the corresponding cumulants – susceptibility, skewness, kurtosis, and Binder cumulant – for each resample. The resulting cumulant vectors are returned directly, without performing any averaging or error estimation.

Arguments

• bin_size::Int: Bin size used for jackknife resampling
• V::Int: Lattice volume, used for normalization of condensate and susceptibility
• obs::Vector{Vector{T}}: List of $Q$-observable vectors per configuration
• w::Vector{T}: Reweighting factors per configuration
• jobid::Union{Nothing, String}: Optional job ID for logging or debugging

Returns

• cumulants_jk::Vector{Vector{T}}: Jackknife-resampled cumulants packed as a vector of vectors:
  1. cond_jk: Chiral condensate
  2. susp_jk: Susceptibility
  3. skew_jk: Skewness
  4. kurt_jk: Kurtosis
  5. bind_jk: Binder cumulant

Each sub-vector has length equal to the number of jackknife bins (n_jk = nconf / bin_size), and can be passed to Deborah.Sarah.Jackknife.jackknife_average_error externally for analysis.

compute_cumulants_raw(
    bin_size::Int, 
    V::Int, 
    obs::Vector{Vector{T}}, 
    jobid::Union{Nothing, String}=nothing
) where T -> Vector{Vector{T}}

Compute raw jackknife-resampled cumulants from single ensemble.

This function performs jackknife resampling of raw $Q$-moments ($Q_{n} \; (n=1,2,3,4)$) and computes the corresponding cumulants – susceptibility, skewness, kurtosis, and Binder cumulant —- for each resample. The resulting cumulant vectors are returned directly, without performing any averaging or error estimation.

Arguments

• bin_size::Int: Bin size used for jackknife resampling
• V::Int: Lattice volume, used for normalization of condensate and susceptibility
• obs::Vector{Vector{T}}: List of raw $Q$-observable vectors per configuration
• jobid::Union{Nothing, String}: Optional job ID for logging or debugging

Returns

• cumulants_jk::Vector{Vector{T}}: Jackknife-resampled cumulants packed as a vector of vectors:
  1. cond_jk: Chiral condensate
  2. susp_jk: Susceptibility
  3. skew_jk: Skewness
  4. kurt_jk: Kurtosis
  5. bind_jk: Binder cumulant

Each sub-vector has length equal to the number of jackknife bins (n_jk = nconf / bin_size), and can be passed to Deborah.Sarah.Jackknife.jackknife_average_error externally for analysis.

compute_moments_raw(
    bin_size::Int,
    obs::Vector{Vector{T}},
    jobid::Union{Nothing, String}=nothing
) where T -> Vector{Vector{T}}

Compute jackknife resamples of the moments $Q_{n} \; (n=1,2,3,4)$ (of single ensemble) from per-configuration inputs obs. For each configuration i, calc_Q is called to obtain (Q1[i], Q2[i], Q3[i], Q4[i]), and then jackknife resampling is applied component-wise.

Arguments

• bin_size::Int: Jackknife bin size.
• obs::Vector{Vector{T}}: Per-configuration vectors consumed by calc_Q.
• jobid: Optional job id for logging.

Returns

• Vector{Vector{T}} with 4 elements: [Q1_jk, Q2_jk, Q3_jk, Q4_jk], where each is a length-njk vector of jackknife resample values.

compute_traces_raw(
    bin_size::Int,
    obs::Vector{Vector{T}},
    jobid::Union{Nothing, String}=nothing
) where T -> Vector{Vector{T}}

Make jackknife resamples for traces $\text{Tr} \, M^{-n} \; (n=1,2,3,4)$ in the same return shape as compute_cumulants_raw, but without computing cumulants.

The input obs[i] is assumed to be a length-5 vector like: [dummy, trM1, trM2, trM3, trM4] The first entry is ignored; only indices 2..5 are used.

Arguments

• bin_size::Int: Jackknife bin size.
• obs::Vector{Vector{T}}: Per-configuration trace vectors (length = 5).
• jobid: Optional job id for logging.

Returns

• Vector{Vector{T}} with 4 elements: [trM1_jk, trM2_jk, trM3_jk, trM4_jk], where each trM*_jk is a vector of length njk containing the jackknife resample values for that trace component.

Notes

• This is a lightweight companion to compute_cumulants_raw. No volume $V$ or $\kappa$-scaling is applied here; it simply jackknifes the provided $\text{Tr} \, M^{-n} \; (n=1,2,3,4)$.