Turning point processes in plane partitions with periodic weights of arbitrary period

Sevak Mkrtchyan

REVIEW 3 major objections 5 minor 21 references

Turning point processes in plane partitions with periodic weights of arbitrary period

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Periodic weights turn a plane partition's single turning point into several correlated GUE-corners processes.

desk verdict New results on turning points for a regularized periodic-weight model, but the claimed universality for true periodic weights is not established because the regularization is visible in the limit. read the letter →

arxiv 1908.01246 v2 pith:6KPBO36U submitted 2019-08-03 math.PR math-phmath.MP

Sevak Mkrtchyan This is my paper · ORCID

classification math.PRmath-phmath.MP MSC 60K3582B2005A17

keywords planepartitionsperiodicweightsturningpointsGUE-cornersprocesslozengetilingsfrozenfacetsSchurdeterminantalpoint

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper studies random plane partitions whose volume weights repeat with period k and asks what the system looks like near its right vertical boundary in the scaling limit. The answer it defends is that a single GUE-corners turning point, familiar from the homogeneous weight case, is replaced by up to k turning points, one for each distinct product of consecutive weights; these turning points are separated by vertical facets whose projections can have arbitrary rational slope. The paper proves that the local point process at each turning point is a GUE-corners process on suitably chosen vertical slices, and that the full collection of turning-point processes is a non-trivial interlacing of several GUE-corners processes. A separate claim is that the weight modification used to make the measure finite creates a first-order phase transition along the corner slice, with a limiting point process invariant only in the vertical direction. If correct, this changes the expected universality of boundary fluctuations for determinantal random tilings under periodic weights.

What carries the argument

The engine is the Okounkov-Reshetikhin double contour-integral formula for the Schur process correlation kernel (equation (3)), whose leading exponential behavior is controlled by the action $S_{\tau,\chi}(z)$ in (8). With $k$-periodic weights, the products $\beta_i=\alpha_{d-1}\cdots\alpha_{d-i}$ collapse to $l$ distinct values $\tilde{\beta}_1<\cdots<\tilde{\beta}_l$ with multiplicities $m_i$; near $\tau=V$ the double real critical points of the action sit at $z=\tilde{\beta}_i e^V+O(\sqrt{\varepsilon})$, and each such critical point is the saddle point of a turning point. The paper regularizes the infinite partition function by multiplying the 0-th slice weight by $\gamma=\prod_{\alpha_j<1}\alpha_j$ (equation (2)); this factor enters the action through the terms $\tilde{\beta}_i\gamma$ and is responsible for the first-order phase transition at the corner slice. Steepest-descent deformation of the $z$- and $w$-contours through $z_j=\tilde{\beta}_j e^V$ produces the limiting kernel (17), whose discrete horizontal and continuous vertical coordinates identify each turning point, slice by slice, with the GUE-corners process.

What would settle it

An independent computation of the $k=3$ or $k=4$ turning-point process in a model with fully periodic weights kept finite by a different boundary modification (for example, a skew-plane-partition cut-out analogous to the $k=2$ construction) would settle the point: if it reproduces the kernel (17) and the rational-slope facets, the locality assumption holds; if it does not, the $\gamma$ modification is not innocuous and Theorem 4.6 describes a model with a defect at the 0-th slice.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.6: as $r\to 0$ near the vertical boundary $\tau=V$, the system develops $l$ turning points, one for each distinct value $\tilde{\beta}_i$ among the products $\beta_i=\alpha_{d-1}\cdots\alpha_{d-i}$; the $j$-th turning point sits at $(V,\chi_j)$ with $\chi_j$ given by (16), and the correlation functions near it are determinants of the kernel (17). Section 4.1 shows that on selected vertical slices this kernel is exactly the GUE-corners kernel: any collection of slices in which the $i$-th slice carries $i$ particles yields the GUE-corners process, so the full turning-point process is several GUE-corners processes interlaced in a non-trivial way. Section 4.2 claims the frozen regions between turning points are vertical facets whose projection has angle $(j-1)\pi/(2k)$, giving arbitrary rational slopes, and each facet is a deterministic periodic pattern of two lozenge orientations. The paper also claims (Theorem 5.1, computed for $k=2$) that the modified weight at the corner slice produces a first-order phase transition with a limiting point process that is translation invariant in the vertical direction but not in the horizontal direction.

Load-bearing premise

The load-bearing premise is that multiplying the weight on the corner slice by $\gamma$ only makes the partition function finite and does not change the turning-point processes, because those points are macroscopically far from that slice; the paper checks this only for $k=2$, and for $k>2$ it is an unproven locality assertion.

Editorial extensions

If this is right

For weights of period $k$, the right boundary hosts up to $k$ distinct turning points, one per distinct consecutive product $\tilde{\beta}_i$, instead of the single turning point of the homogeneous model.
Each turning point carries a GUE-corners process on any family of vertical slices whose $i$-th slice has $i$ particles, so the boundary process is a non-trivial interlacing of several GUE-corners processes.
The frozen regions between turning points are vertical facets whose projection angle can be any rational multiple $(j-1)\pi/(2k)$, realized by periodic deterministic patterns of two lozenge orientations.
In the bulk near the edge, the limiting correlation kernel is a product over the distinct $\tilde{\beta}_i$'s, recovering the incomplete-beta kernel in the homogeneous case and breaking horizontal translation invariance to translations by $k\mathbb{Z}\times\mathbb{Z}$.
The weight modification at the corner slice induces a first-order phase transition whose limiting point process (computed for $k=2$) is translation invariant vertically but not horizontally.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: if the locality assumption holds for every $k$, the number and type of turning-point processes is determined entirely by the combinatorial data $(\tilde{\beta}_i,m_i)$, so the boundary universality class is indexed by the weight sequence's distinct consecutive products rather than by its period alone.
Inference: a numerical check for $k>2$ — exact sampling or direct kernel evaluation with the regularization placed at different slices far from the right boundary — should leave the turning-point kernel unchanged; agreement would confirm locality, disagreement would reveal that the defect at the 0-th slice reaches the boundary.
Inference: the vertical-facet construction suggests that other dimer models with periodic weights and a Newton polygon carrying several boundary lattice points will exhibit analogous countable families of tilted frozen facets, with the present rational-slope mechanism as a template.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—

Here's the short version: this is a genuinely new piece of work on turning points in plane partitions with periodic weights, but the main universality claim rests on an unproven locality assumption. The paper proves its theorems for a regularized model with a modified weight at slice 0, and the modification is not invisible in the limit: γ shows up in the leading-order action (8), in the turning-point heights (16), and in the kernel (17). For k>2 there is no argument that the γ-dependent limit equals what the true periodic model would produce if it existed. The k=2 check against [Mkr14] is encouraging, but it doesn't establish the general claim.

What's actually new: arbitrary-period turning point analysis with several correlated GUE-corners processes, vertical facets with rational slopes, and an explicit phase-transition kernel in the k=2 case. The saddle point analysis is standard, but the phenomena are new. The paper is careful with theorem statements, and the derivation of the kernel is mostly explicit. The description of the frozen-region profiles (Section 4.2) is a nice touch.

Soft spots, in order of size. The γ-regularization is the big one. The authors state that the modification 'should have no effect' on turning points because they are macroscopically far from slice 0, but the action integral over M∈(−∞,0) contributes at leading order, so the turning point locations and the local kernel depend on γ. That is a real gap. It doesn't destroy the paper: the theorems are correctly proven for the regularized model. But it means the paper's title and abstract overclaim, and the reader should treat 'periodic weights of arbitrary period' as 'a specific regularized version of periodic weights.' The second soft spot is minor: the phase transition result is proven only for k=2 and V=∞, while the abstract presents it without that qualification. A revised version should either prove γ-independence or clearly frame the results as being about the regularized model.

Citation pattern looks fine; the relevant literature is covered, and the self-citations are for results used directly.

Who it's for: specialists in random plane partitions and dimer models. It deserves a serious referee. I would send it to review, but ask the referee to focus on whether the γ-dependence can be removed or whether the results should be reframed. If the authors do that, the paper becomes a solid contribution. As it stands, I'd give it conditional, not unconditional, acceptance.

Referee Report

3 major / 5 minor

Summary. The paper studies random plane partitions with weights that are periodic of period k in the slice variable, in the scaling limit r→0 with the size of the box growing as V/r. Because the unmodified periodic weights make the partition function infinite, the author multiplies the weight at the 0-th slice by a factor γ (Eq. (2)), producing a model that is periodic except at one slice. Using the determinantal Schur process kernel of Okounkov–Reshetikhin, the paper derives leading asymptotics of the kernel (Eq. (8)), locates turning points near the vertical boundary τ=V (with heights χ_j in Eq. (16)), and computes the limiting correlation kernels in the bulk near the edge (Theorem 4.2) and near the turning points (Theorem 4.6). It then argues that each turning point yields a GUE-corners process on selected vertical slices (Section 4.1) and that the frozen regions between turning points are vertical facets with arbitrary rational slope (Section 4.2). It also analyzes a first-order phase transition at τ=0 in the k=2 case (Theorem 5.1).

Significance. If correct, the main results establish a new family of turning-point universality classes: several GUE-corners processes that are nontrivially correlated, together with frozen facets of arbitrary rational slope. The paper is careful in stating precise theorems with explicit contour-integral kernels, and the k=2 comparison with the author's earlier work [Mkr14] is a useful consistency check. However, the central claims are formally about the regularized model with a defect at slice 0, and the assertion that this defect does not affect the turning points is not proven for k>2. Thus the significance for genuinely periodic weights is conditional, and the paper would benefit from either a proof of locality or a reframing of the claims.

major comments (3)

[1 (p.4), Eq. (2), (8), (16), (17)] The regularization at slice 0 is not proven to be invisible for the turning-point processes. The paper states (p.4) that multiplying q_0 by γ 'should have no effect' because the turning points are macroscopically far away, and the only evidence is the k=2 comparison with [Mkr14]. However, γ appears explicitly in the leading-order action (8) through the Φ^- term integrated over M<0, in the turning-point heights χ_j in (16), and in the kernel (17) via S''. Since the unmodified periodic weights do not define a probability measure (as shown in Section 2), Theorems 4.2 and 4.6 are theorems about a model with a line defect at τ=0. This is load-bearing because the abstract and introduction claim results for 'periodic weights of arbitrarily high period'. The author should either provide a proof that the turning-point asymptotics are independent of the regularization (for instance, by showing that the γ-dependence cancels to the relevant order), or reframe the theorems and abstract as applying to the regularized model.
[4.0.2, Eq. (13)] The step 'By setting Sp(zj,±)=0 and solving for χ from (9) we obtain' (Eq. (13)) is too compressed for a result that determines the turning-point heights used in Theorem 4.6 and Section 4.1. In particular, the text says that the O(ε) coefficient g_ε in zj,± cancels, but the reader cannot verify that no √ε contributions are lost without seeing the full expansion. Please expand the derivation or include an appendix with the intermediate algebra.
[4.2] The claim that the frozen regions between turning points have arbitrary rational slope rests on the assertion 'It can inductively be shown, that after "passing through" a turning point, exactly one of the L's in the repeating pattern changes to an R.' This is not proved, and the section is written as a heuristic discussion rather than a formal theorem. Since arbitrary rational slope is prominently advertised in the abstract, this inductive argument should be made rigorous or explicitly labeled as a conjecture/consequence.

minor comments (5)

[4.0.2] There are two typos: 'z_{2,-1}' should be 'z_{2,-}' and 'χ_{2,-1}' should be 'χ_{2,-}' (appearing in the inequalities after Eq. (13)).
[Remark 4.4] 'decay exponentially fast as r→∞' should read 'as r→0', since the scaling limit throughout is r→0; as written it is inconsistent with the rest of the paper.
[4.0.2] The notation θ(1) with the parenthetical 'means of constant order as ε→0' is nonstandard; consider using Θ(1) or writing explicit bounds instead.
[2] In the discussion of ill-definedness of the measure, the phrase 'the measure P_bar_q is not well-defined' would be clearer if it read 'the partition function is infinite, so the normalized probability measure does not exist'.
[Abstract and Introduction] The abstract's claim about 'periodic weights of arbitrarily high period' is misleading until the regularization is discussed; please add a sentence in the abstract noting that a weight modification at one slice is used.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the turning-point and facet results are a self-contained saddle-point analysis of the explicit kernel from [OR07]; no fitted parameter is renamed as a prediction.

full rationale

The derivation chain is: define the measure by weights (2) (model inputs α_i, γ, k, V); invoke the determinantal kernel of the Schur process, Theorem 3.1 from [OR07] (external); compute the leading-order action S_{τ,χ} in (8); locate double real critical points; derive turning-point heights χ_j in (16); expand the kernel near each critical point to obtain the limiting kernels (17) and (18); and derive frozen-region profiles from the counts N_{t,i}. Each step is a computation from the stated model, and the limiting kernels are not fitted to any subset of data. The identification of the slice-restricted kernel (18) as GUE-corners is a mathematical identification, not a definitional restatement. The author's own citations are not load-bearing circularity: Lemma 4.1 from [Mkr11] is quoted with its hypotheses and is a standalone complex-angle lemma, and the [Mkr14] comparison is presented as a consistency check for k=2, not as the derivation of the turning-point results. The only caveat, which is a correctness/interpretation risk rather than circularity, is the stated assumption (Section 1, p.4 and Section 2) that multiplying q_0 by γ in (2) has no effect on turning points because they are 'macroscopically far away' from slice 0; the paper verifies this only for k=2, and γ explicitly enters (8), (16), and (17). Since the theorem statements expressly concern the weights (2), the proofs are self-contained for that model; the unproven locality assertion concerns whether those theorems describe the unmodified periodic weights, which are not defined because their partition function diverges. That gap does not make any equation equal to its own input, so the circularity score remains 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The model is defined by the periodic weights α_0,...,α_{k-1} with product 1 and a fixing parameter V, which are quantified variables in the theorems, not fitted constants. The main derivation relies on the known determinantal kernel for the Schur process (OR07), standard saddle point analysis, and a model-specific assumption that the weight modification at the 0-th slice is harmless for the turning point statistics.

assumptions (3)

domain assumption The Schur process correlation functions are determinantal with kernel (3) (Theorem 3.1 from [OR07]).
This imported theorem is the starting point for all asymptotic calculations in Sections 4 and 5.
domain assumption Multiplying q_0 by γ (Eq. (2)) makes the measure well-defined and does not change the processes at turning points.
Load-bearing modeling assumption stated on page 4; checked only for k=2 via [Mkr14].
standard math The leading asymptotics of the kernel are governed by steepest descent through critical points of S_{τ,χ}(z); the deformed contours pick up only the stated residues.
Used in Sections 4.0.4, 4.0.5 and 5; a standard analytic technique whose validity is assumed for the stated contour choices.

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Pith. "Pith review of Turning point processes in plane partitions with periodic weights of arbitrary period." pith.science (2026). https://pith.science/paper/6KPBO36U

@misc{pith2026190801246,
  author       = {Pith},
  title        = {Pith review of: Turning point processes in plane partitions with periodic weights of arbitrary period},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6KPBO36U}},
  note         = {Machine review of arXiv:1908.01246}
}

read the original abstract

We study random plane partitions with respect to volume measures with periodic weights of arbitrarily high period. We show that near the vertical boundary the system develops up to as many turning points as the period of the weights, and that these turning points are separated by vertical facets which can have arbitrary rational slope. In the lozenge tiling formulation of the model the facets consist of only two types of lozenges arranged in arbitrary periodic deterministic patterns. We compute the correlation functions near turning points and show that the point processes at the turning points can be described as several GUE-corners processes which are non-trivially correlated. The weights we study introduce a first order phase transition in the system. We compute the limiting correlation functions near this phase transition and obtain a process which is translation invariant in the vertical direction but not the horizontal.

Figures

Figures reproduced from arXiv: 1908.01246 by the authors.

**Figure 1.** A plane partition π and the corresponding stack of cubes/lozenge tiling 0 0 0 0 0 0 0 2 0 0 0 0 0 1 2 3 0 0 0 1 2 3 3 3 0 0 1 2 2 3 4 4 1 2 3 3 4 5 6 6 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** All possible frozen regions of slope 0 that can arise when weights are 6-periodic. In the formulation of the model as a dimer model the weights we take essentially correspond to taking a larger fundamental domain in the language of [Ken09] (see [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The smaller region is the fundamental domain when weights are 4-periodic. b α1 c α2 α3 α4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: The smaller region is the fundamental domain when weights are homogeneous. a b c When dealing with the measure (1) with appropriate non-homogeneous weights care should be taken to ensure that the measure is indeed well-defined, as the partition function becomes infinit…

**Figure 5.** Figure 5: Coordinate axes h t the limit qt = q → 1 −. In the periodic case one could consider the limit when qi → 1 −, ∀i, however in that regime the behavior of the system is the same as that of the homogeneous measure with the weight given by the geometric average of the perio…

**Figure 6.** Figure 6: The configuration π m when αk−1 < 1 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 7.** Figure 7: The configuration π m when αk−1 > 1. Lastly, if αk−1 = 1, then we can consider αk−2, and so on. This shows, that unless α0 = · · · = αk−1 = 1, the measure Pq¯ is not well defined if q is close enough to 1. One way to fix the issue is to modify the boundary. This was th…

**Figure 8.** Figure 8: Weights assignments of period 4. The number in a cell indicates the weight of the boxes above the cell. α4 γ α3 α2 α1 α4 α3 α2 α1 α1 α4 γ α3 α2 α1 α4 α3 α2 α2 α1 α4 γ α3 α2 α1 α4 α3 α3 α2 α1 α4 γ α3 α2 α1 α4 α4 α3 α2 α1 α4 γ α3 α2 α1 α1 α4 α3 α2 α1 α4 γ α3 α2 α2 α1 α4 …

**Figure 9.** Figure 9: The integration contours and the intervals where there cannot be real critical points. 0 β˜1γ β˜2γ β˜ . . . lγ β˜1eτ β˜1eV . . . β˜leτ β˜leV Cz Cw Note also, that the Cz contour from (3) contains no zeros of Φ+ q¯ (z, t1), i.e. none of e r(m+ 1 2 )β˜ i for t1 < m < d, …

**Figure 10.** Figure 10: A plot of the function Sp(z). β˜2e ˜ V 0 βlγ β˜1eτ β˜1eV β˜ . . . 2eτ β˜leτ β˜leV χ1,− − χ χ1,+ − χ χ2,− − χ χ2,+ − χ χl,− − χ χl,+ − χ −V − χ + τ 2 . . . 4.0.3. The number of real and complex critical points. First, we count the critical points of Sτ,χ when χ is larg…

**Figure 11.** Figure 11: Setup of Lemma 4.1 [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 12.** Figure 12: The shaded region is the region <(Sτ,χ(z)) > <(Sτ,χ(zcr)). The solid curves are the original contours whereas the dotted ones are the new contours. 0 β˜j eτ β˜1e β˜ τ lγ β˜j eV Cz Cw C� z C� w Since t1, t2 ≈ τ r > 0, from (4) we have Φq¯(z, t1) Φq¯(z, t2) = Φ − q¯ (z,…

**Figure 13.** Figure 13: The shaded region is the region <(SV,χj (z)) > <(SV,χj (zj )). The solid curves are the original contours whereas the dotted ones are the new, deformed contours. 0 β˜j eV Cw C� z C� w Cz β˜kγ β˜1eV [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]

**Figure 14.** Figure 14: The ξ and ω contours. the change of variable z = zje r 1/2 ξ , w = zje r 1/2ω . In these coordinates the steepest descent contours are as in [PITH_FULL_IMAGE:figures/full_fig_p016_14.png]

**Figure 15.** Figure 15: The frozen regions corresponding to the profiles (19). The l < k case is similar, with the only difference being that some facets are missing. For example, if k = 4 and β1 = β2 > β3 > β4, then there will only be 3 turning points and the (L 3R) facet will be missing as…

**Figure 16.** Figure 16: An exact sample(generated using the algorithm in [Bor11]) of plane partitions with 3-periodic weights, modified at the slice through the corner. Near the right edge of the figure the 3 turning points and the frozen regions separating them are visible. 5. Point process…

**Figure 17.** Figure 17: The sequence ym for m ∈ Z + 1 2 . 1 2 3 2 5 2 7 2 9 2 11 2 −1 2 −3 2 −5 2 −7 2 −9 2 −11 2 11 1 1 11 1 α 1 α 1 α 1 α 1 α 1 α m ym From (9) and (10) for τ ≥ 0 we have (22) 2z dSτ,χ dz (z) = − ln z − 1 z − ln z − 1 α z + ln −1 z − e τ + ln −α z − αeτ − 2χ…

**Figure 18.** Figure 18: A plot of the curve χ±(τ ) when a = 9. τ χ sign of τ we have (27) χ±(τ ) = − ln(1 ∓ e − |τ| 2 ) − ln(1 ∓ α −1 e − |τ| 2 ) − 1 2 |τ | [PITH_FULL_IMAGE:figures/full_fig_p022_18.png]

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Correlation functions for determinantal processes defined by infinite block Toeplitz minors

T. Berggren and M. Duits . Correlation functions for determinantal processes defined by infinite block Toeplitz minors . arXiv e-prints , page arXiv:1901.10877, Jan 2019

work page Pith review arXiv 1901

[2] [2]

Random skew plane partitions with a piecewise periodic back wall

Cedric Boutillier, Sevak Mkrtchyan, Nicolai Reshetikhin, and Peter Tingley. Random skew plane partitions with a piecewise periodic back wall. Ann. Henri Poincar\'e , 13(2):271--296, 2012

work page 2012

[3] [3]

Periodic S chur process and cylindric partitions

Alexei Borodin. Periodic S chur process and cylindric partitions. Duke Math. J. , 140(3):391--468, 2007

work page 2007

[4] [4]

Schur dynamics of the S chur processes

Alexei Borodin. Schur dynamics of the S chur processes. Adv. Math. , 228(4):2268--2291, 2011

work page 2011

[5] [5]

Gibbs Ensembles of Nonintersecting Paths

Alexei Borodin and Senya Shlosman . Gibbs Ensembles of Nonintersecting Paths . Communications in Mathematical Physics , 293(1):145--170, Jan 2010

work page 2010

[6] [6]

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Sunil Chhita and Kurt Johansson. Domino statistics of the two-periodic aztec diamond. Advances in Mathematics , 294:37 -- 149, 2016

work page 2016

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Cesar Cuenca . Universal behavior of the corners of Orbital Beta Processes . arXiv e-prints , page arXiv:1807.06134, Jul 2018

work page Pith review arXiv 2018

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Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory

Vadim Gorin and Greta Panova. Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. Ann. Probab. , 43(6):3052--3132, 11 2015

work page 2015