bibkey: ptaszynski2022skewness
authors: Krzysztof Ptaszyński
year: 2022
title: "Bounds on skewness and kurtosis of steady-state currents"
doi: 10.1103/PhysRevE.106.024119
url: https://arxiv.org/abs/2204.08421v4
claim: "The paper derives bounds on the skewness and kurtosis of steady-state currents for several classes of systems; for unicyclic Markovian networks it infers numerically that the extremal values are attained when all clockwise rates are equal but one, and on that topology finds the minimum skewness -(8-8N+N^2)/(16(N-1)^2) of a unidirectional N-state cycle at a = 1/(N-1); in the limit N -> infinity it gives the minimum kurtosis -(1+sqrt(5))/10 for a = A/N, A = (3-sqrt(5))/2, and the minimum of skewness plus kurtosis -8/27 for A = 1/2; it lists the product of skewness and kurtosis in [-0.054, 1] (eq. produni), the lower value approximate."
strata_touched:
  - D5/S3/StatisticalMechanics/UnicyclicSkewnessMinimum
  - D5/S3/StatisticalMechanics/UnicyclicKurtosisLowerBound
  - D5/S3/StatisticalMechanics/UnicyclicSkewnessPlusKurtosisLowerBound
  - D5/S3/StatisticalMechanics/UnicyclicSkewnessMinusKurtosisUpperBound
  - D5/S3/StatisticalMechanics/UnicyclicSkewnessMinusKurtosisLowerBound
  - D5/S3/StatisticalMechanics/UnicyclicSkewnessTimesKurtosisLowerBound
license: citation-only
triage: anchor

Ptaszyński, bounds on skewness and kurtosis of steady-state currents

K. Ptaszyński, arXiv:2204.08421v4 [cond-mat.mes-hall] (20 July 2022; v1 18 April 2022); Phys. Rev. E 106, 024119 (2022). Quotations are from the v4 TeX source, with the macros \llangle, \rrangle written as $⟨⟨$, $⟩⟩$ and cross-references omitted.

The skewness of a current:

$\mathcal{S}=\frac{⟨⟨j^3⟩⟩}{⟨⟨j^1⟩⟩}$,

with the scaled cumulants $⟨⟨j^n⟩⟩$. For a unicyclic network the winding-number cumulants follow from the waiting-time cumulants (Appendix):

$⟨⟨j^1⟩⟩=\frac{1}{{\kappa }_1}$, $⟨⟨j^3⟩⟩=3\frac{{\kappa }_2^2}{{\kappa }_1^5}-\frac{{\kappa }_3}{{\kappa }_1^4}$, with ${\kappa }_1={\sum }_{i=1}^N{k}_{i+1,i}^{-1}$, ${\kappa }_2={\sum }_{i=1}^N{k}_{i+1,i}^{-2}$, ${\kappa }_3=2{\sum }_{i=1}^N{k}_{i+1,i}^{-3}$

for clockwise rates $k_{i+1,i}$ ($k_{N+1,N}=k_{1N}$). The unidirectional minimum and the conjectured extremal topology (Section III E 1):

it was inferred that the bounds are always saturated for a specific type of network topology with $k_{21}=ak$ and $k_{ij}=k$ for $j>1$ … Assuming this type of network topology, the value of $a$ saturating the bounds has been then analytically determined: $\min (\mathcal{S})=-\frac{8-8N+N^2}{16(N-1{)}^2}$ for $a=\frac{1}{N-1}$, $\max (\mathcal{S})=1$ for $a\to 0$.

The kurtosis is $\mathcal{K}=⟨⟨j^4⟩⟩/⟨⟨j^2⟩⟩$ (l. 54). In the same section the remaining unidirectional bounds are obtained asymptotically:

The other bounds have been derived in the assymptotic limit $N\to \infty$ in which the saturating value of $a$ scales as $a=A/N$. They read as $\min (\mathcal{K})=-\frac{1+\sqrt{5}}{10}$ for $A=\frac{3-\sqrt{5}}{2}$, $\max (\mathcal{K})=1$ for $A\to 0$, …

and, for the sum (l. 434–437):

$\min (\mathcal{S}+\mathcal{K})=-\frac{8}{27}$ for $A=\frac{1}{2}$, $\max (\mathcal{S}+\mathcal{K})=2$ for $A\to 0$

it was numerically conjectured that the minimum and maximum values of $\mathcal{S}$, $\mathcal{K}$, $\mathcal{S}+\mathcal{K}$, $\mathcal{S}-\mathcal{K}$ and $\mathcal{S}\times \mathcal{K}$ are always obtained for a specific type of network topology, with all rates $k_{ij}$ equal to $k$ apart from one (here $k_{21}$) equal to $ak$.

Verified locator

Source: https://arxiv.org/abs/2204.08421v4

• DOI: https://doi.org/10.1103/PhysRevE.106.024119 (Phys. Rev. E 106, 024119 (2022); author, journal, volume and article number from Crossref).
• URL: https://arxiv.org/abs/2204.08421v4 (the e-print is a gzipped tar archive; member main.tex, 1102 lines, md5 5b72a7d5de1e5c34c76db21badb56492):
  • the skewness, l. 53;
  • unidirectional networks, the skewness minimum and the asymptotic kurtosis minimum, Section III E 1, l. 407–432;
  • the asymptotic minimum of $\mathcal{S}+\mathcal{K}$, l. 434–437;
  • the winding-number cumulants and the waiting-time cumulants, Appendix, l. 768–790;
  • the numerically conjectured extremal topology, Appendix, l. 792.
• The author and the arXiv dates are from the arXiv metadata.

Exact skewness-minus-kurtosis upper endpoint and source sign

The primary v4 source main.tex, SHA-256 0d8f9bf311a07990b64d4746e9ec4fb1e164deec23b270503642b94766baf84e, line 432, states

$\max (\mathcal{S}-\mathcal{K})=3(107+51\sqrt{17})/2048$ for $A=(5-\sqrt{17})/4$.

Figure 6 also concerns $\mathcal{S}-\mathcal{K}$. In contrast, the summary at line 133 prints $\mathcal{K}-\mathcal{S}\in [-0.15,0.465]$. These signs conflict. The difference endpoint formalization follows the detailed display and figure; it does not silently read the summary as the same formula. Line 424 uses $a=A/N$ in the infinite-size limit, and line 792 labels the general extremal-topology assertion numerical/conjectural. The source’s analytic evaluation on that topology is credited input; the universal inference over arbitrary positive rate vectors requires proof.

The explicit finite realization with inverse rates $u$ and $N-1$ copies of $(1-u)/(N-1)$, where $u=(9+\sqrt{17})/16$, has relative exceptional rate $A/(N-1)$, not $A/N$, with $A=(1-u)/u=(5-\sqrt{17})/4$. The ratio of the two relative rates is $N/(N-1)$ and tends to one. Both finite families have positive rates; equality of their asymptotic scaling does not identify their finite members.

The formal domain retains arbitrary positive real coordinates and repeated rates. At $N=2$ its current interpretation uses two oriented phase stages and a marked winding crossing. The formal expressions do not construct the process, prove renewal asymptotics, or assert a physical $N=1$ case. The independently inspected literature scopes found no exact supplier; this is not a worldwide priority claim. The journal body and some citation-list memberships remain unverified.

Exact skewness-minus-kurtosis lower endpoint

The same primary v4 source and hash, main.tex line 431, state

$\min (\mathcal{S}-\mathcal{K})=3(107-51\sqrt{17})/2048$ for $A=(5+\sqrt{17})/4$.

The detailed display and Figure 6 determine the sign used here; the conflicting summary at line 133 remains as described above. The asserted lower endpoint concerns the same scaled current cumulant ratios and the same numerical extremal-topology conjecture. Its universal positive-rate inequality and greatest-lower-bound interpretation do not assume that conjectured topology.

For the explicit finite approximation with inverse rates $\alpha$ and $N-1$ copies of $(1-\alpha )/(N-1)$, take $\alpha =(9-\sqrt{17})/16$. Its relative exceptional rate is $A/(N-1)$ with $A=(1-\alpha )/\alpha =(5+\sqrt{17})/4$. This is a directly evaluated positive finite family; it is not identified with the source’s finite $A/N$ family. The domain and physical-interpretation limits above also apply to this lower endpoint.