Complex Langevin Simulations and Zeroes of the Measure. I.-O. - - PowerPoint PPT Presentation

Complex Langevin Simulations and Zeroes of the Measure.

I.-O. Stamatescu (Heidelberg) Results in the frame of common work with:

• G. Aarts (Swansea), E. Seiler (Munich) and D. Sexty (Wuppertal)

and further collaboration with

• F. Attanasio, L. Bongiovanni, B. J¨

ager (Swansea) and J. Pawlowski (Heidelberg). SIGN 2015, Debrecen

1

Items of the discussion

• 1. CLE: setup and problems.
• 2. Discussion of the formal proof of equivalence.
• 3. Dealing with the non-holomorphy in an effective model.
• 4. Taking over some conclusions to HD-QCD.
• 5. Discussion.

Discussion and results are part of the analysis in an upcoming paper by the mentioned authors.

2

• 1. CLE set up and problems

3

Complex, holomorphic action − → complex drift K(z) = −∇zS, − → imaginary parts for the variables − → Process on the complex extension of the original manifold: z(t) = x(t) + i y(t) , x ∈ Mr , z ∈ Mc The process realises a positive probability distribution P(x, y). Formal equivalence theorem: for analytic observables O(x, y) the averages over the process reproduce the ensemble averages with the (complex!) distribution ρ(x) = exp(−S(x)): what we get: OP (t) = Oρ(t) : what we want (t → ∞)

• Cf. G. Aarts, E. Seiler and I. -O. Stamatescu, Phys. Rev. D 81

(2010) 054508 [arXiv:0912.3360 [hep-lat]].

• G. Aarts, F. A. James, E. Seiler and I. -O. Stamatescu, Eur. Phys.
• J. C 71 (2011) 1756 [arXiv:1101.3270], etc.

4

As with every numerical method the the proof of equivalence relies on certain conditions to be fulfilled, for CLE in particular: 1 holomorphy of the drift and of the observables, 2 sufficient fall off of P(x, y) in the y−direction. There are also ”Practical problems”:

• 3. Accumulation of numerical errors. Typical effect: run-aways,

divergence of some quantities. K(z) becomes unbounded

• 4. Unprecise sampling - in the presence of trajectories going far in the

y direction, a further effect of 2 above.. Require adaptive step size, controlling the distribution, etc. Notice: there are many processes K(z) (P(x, y)) leading formally to the desired EV’s. This can be used in controlling the method.

5

• 2. Discussion of the Formal Proof of Equivalence

6

Formal proof Consider first holomorphic drift. P(x, y) and the observables evolve with the Fokker-Planck equation and its adjoint: ˙ P(x, y; t) = LT P(x, y; t), (1) ˙ O(x + iy; t) = L O(x + iy; t) = ˜ L O(z; t), (2) L = (NR∇x + Kx)∇x + (NI∇y + Ky)∇y (3) LT = ∇x [NR∇x − Kx] + ∇y [NI∇y − Ky] , (4) ˜ L = [∇z − (∇zS(z))] ∇z, (z = x + iy, NR − NI = 1) (5) K(z) = −∇zS(z) = ρ(z)−1 ∇zρ(z), ρ(z) = e−S(z) (6) Here we used the Cauchy-Riemann eqs. We shall set NR = 1, NI = 0.

7

Consider the interpolation function: F(t, τ) ≡

• P(x, y; t − τ)O(x + iy; τ)dxdy ,

(7) F(t, 0) =

• P(x, y; t)O(x + iy)dx dy = OP (t),

(8) F(t, t) =

• O(x; 0)ρ(x; t)dx = Oρ(t)

(9) Equivalence is ensured if ∂ ∂τ F(t, τ) = − LT P(x, y; t − τ)

• O(x + iy; τ)dxdy

(10) +

• P(x, y; t − τ)LO(x + iy; τ)dxdy = 0

(11) which follows with integration by parts if there are no boundary terms.

8

Boundary terms from large |z| (cf. Condition 1!) Typically x is compact, thus for holomorphic drift K boundary terms may only come from the non-compact direction y. For gauge-theories our method of Gauge Cooling efficiently concentrates the distribution around the compact domain! (which also helps with the practical problems)

• Cf. E. Seiler, D. Sexty and I. -O. Stamatescu, Phys. Lett. B 723,

213 (2013) [arXiv:1211.3709 [hep-lat]].

9

Meromorphy (cf. Condition 2!) Zeroes of ρ(x) (e.g. from fermionic determinants, first stressed by Molgaard and Splittorff) will lead to a meromorphic drift. The following discussion is due to E. Seiler. To define a holomorphy domain we can introduce further boundaries cutting small circles of radius ǫp around the poles zp. In all examples we find P(xp, yp) = 0. Then we can take the limit ǫp → 0 if the evolution of the (originally analytic) observables O(z; t) = exp(˜ Lt)O(z) =

∞

• n=0

tk k! ˜ LkO (12) does not introduce essential singularities. This can be shown to hold in special cases of second order zeroes.

10

Example: −S(z) = −ωz2 2 + ln(zn), K(z) = n z − ωz, (13) ˜ L = d2 dz2 + n z − ωz d dz (14) then it easy to see that ˜ Lz = n z − ω, ˜ L2z = n(2 − n) z3 + ω2z, ˜ L1 z = 2 − n z3 + ω z , · · · (15) and for n = 2 no higher singularities are produced! (This example can be solved, e.g. for O(z) = z2, n = 2 we obtain et˜

Lz2 = e−2ωt

• z2 − 3

ω

• + 3

ω − → 3 ω = Oρ (t → ∞) (16) which is the exact result.)

11

• 3. The one link SU(3) effective model

One link in the field of its neighbors: Seen as one Polyakov line of a 4-dim lattice model in temporal gauge

12

Effective model for HD-QCD −S = β

3

• i=1
• eαiei wi + e−αie−i wi

+ ln Det + ln H (17) H = sin2 w2 − w3 2 sin2 w3 − w1 2 sin2 w1 − w2 2 (18) On = tr(U n) = ei nw1 + ei nw2 + ei nw3, w1 + w2 + w3 = 0, (19) K = −∇S (20) The matrices A = {eαi} ∈ GL(3, C) simulate the staples (after diagonalisation) The process runs in three or (equivalently) two angles, with correspondingly three (two) noise terms.

13

The determinant is Det = (D ˜ D)2 (21) D = 1 + CtrU + C2trU −1 + C3 = (1 + C3)(1 + a P + b P ′), (22) ˜ D = 1 + ˜ CtrU −1 + ˜ C2trU + ˜ C3 = (1 + ˜ C3)(1 + ˜ a P ′ + ˜ b P), (23) a = 3C 1 + C3 , b = C a, ˜ a = 3 ˜ C 1 + ˜ C3 , ˜ b = ˜ C ˜ a, (24) C = 2κeµ, ˜ C = 2κe−µ, P = 1 3trU, P ′ = 1 3trU −1 (25) The square corresponds to dimension 4 of the simulated lattice HD-QCD model, a, b have maxima at C = 2−1/3 and 21/3, respectively with the same value 22/3 independently on C.

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• 0.5

0.5 1 1.5 2 2.5 0.5 1 1.5 2 SU3, 2 angles, kappa=5.5E-6, mu=0, vs beta: O1 ex O-1 ex O2 ex O-2 ex 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.5 1 1.5 2 2.5 3 3*x/(1+x**3) 3*x**2/(1+x**3)

Figure 1: Observables vs β at µ = 0 (left plot). The coefficients a = 3 C/(1 + C3), b = 3 C2/(1 + C3) vs C (right plot). We shall use β = 0.25, 0.75, 1.25. a, b (˜ a, ˜ b) solely depend on C ( ˜ C).

15

Setup as effective model to simulate an N 3

σNτ lattice model.

The relevant parameters are C, ˜ C and thus we should choose: 2 κ = (2 κNτ )Nτ , µ = Nτ µNτ (26) For reference we take Nτ = 8, 16. We shall work at κNτ = 0.12.

Figure 2: a, b at κNτ = 0.12 vs µNτ for Nτ = 8 and 16.

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Typical Behaviour

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 \beta=0.5, \kappaL=0.12, vs \muL: a b O1 ex O-1 ex O2 ex O-2 ex O3 ex O-3 ex O1 O-1 O2 O-2 O3 O-3 0.2804

• 0.2412

0.9311

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.41 1.415 1.42 1.425 1.43 1.435 1.44 1.445 \beta=0.5, \kappaL=0.12, no cut, vs \muL: O1 ex O-1 ex O2 ex O-2 ex O3 ex O-3 ex

Figure 3: κNτ = 0.12, β = 0.25, ordered lattice αi = 0: observables and 1

3a, 1 3b vs µNτ (left), and zoom of the ”dangerous” region (right),

tLangevin ∼ 200.

17

• 250
• 200
• 150
• 100
• 50

• log(min|D|)

• 250
• 200
• 150
• 100
• 50

• log(min|D|)

• 250
• 200
• 150
• 100
• 50

• 1
• 0.5

• 0.1
• 0.05

• 10
• 5

Figure 4:

β = 0.25, κNτ = 0.12, αi = 0, Trajectory analysis and Det scatter plots: µNτ = 1.375, dmin > 10−4; µNτ = 1.425, dmin ∼ 10−9; µNτ = 1.475, dmin > 10−4, dmin ≡ min|Det |. Red points from traj. with dmin > dc = 10−6, Blue from dmin < dc.

Observe clear correlation!:

• trajectories with dmin > dc = 10−6 allways produce correct results,
• wrong results only come from trajectories with dmin < dc.

18

Separate behaviour, fish and whiskers

• 0.04
• 0.02

0.02 0.04

• 1

1 2 3 4 5 6 7 8

• 1

1 2 3 4 5 6 7 8 2000 4000 6000 8000 10000

Figure 5: β = 0.25, κNτ = 0.12. αi = 0. Left: Scatter plots of the unsquared determinant D ˜ D at µ = 1.425, from trajectories with dmin > dc (red fish) and with dmin < dc (blue whiskers), dc = 10−6. Right: Long Langevin time history of Re (D ˜ D), blue lines are trajectories with blue points - typically present only in the thermalization phase. The red fish trajectories lead to the correct results, the blue whiskers to the wrong ones!

19

The cut off dc needs no fine tuning

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.41 1.415 1.42 1.425 1.43 1.435 1.44 \beta=0.5, \kappaL=0.12, |D|<1E-6, vs \muL: O1 ex O-1 ex O2 ex O-2 ex O3 ex O-3 ex

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 \beta=0.5,\, \kappaL=0.12, \mu=1.425, |D|>cut, vs cut: O1 ex O-1 ex O2 ex O-2 ex O3 ex O-3 ex O1 O-1 O2 O-2 O3 O-3

Figure 6: β = 0.25, κNτ = 0.12, αi = 0, observables vs µNτ from trajectories with dmin > dc = 10−6 (left plot) and dependence of the

• bservables on dc at µNτ = 1.425 (right plot).

20

Transient behaviour:

trU trU −1 trU 2 trU −2 tlang CL (0.885,-0.009) (0.892, 0.009) (-0.597, 0.002) (-0.595,-0.002) 600+2800 CLD (1.069, 0.000) (1.073, 0.000) (-0.648,-0.000) (-0.640,-0.002) 600+2800 CLR (1.069, 0.000) (1.072, 0.000) (-0.649,-0.001) (-0.641,-0.001) 1000+5500 CL long (1.063,-0.017) (1.067,-0.002) (-0.648, 0.002) (-0.640,-0.003) 0+11000 CLD long (1.068, 0.000) (1.072, 0.001) (-0.648,-0.000) (-0.640, 0.003) 0+11000 exact (1.069, 0.000) (1.073, 0.000) (-0.649,-0.000) (-0.640, 0.000)

Table 1: Simulation results at β = 0.25 , κNτ = 0.12 , µNτ = 1.425, αi = 0: all trajectories (CL), trajectories with dmin > dc (CLD), all trajectories after dropping the points with Re D < 0 (CLR), long runs without thermalization CL long and CLD long. The deleted Langevin time for the CLR data is typically 0.1%.

21

The effect of the neighbours Disorder appears to increase the metastability of the transient but do not qualitatively change the picture. Strong disorder, however, can lead to results differing from the correct ones by up to a few percents for higher observables (suggesting possible effects from skirts).

• 0.15
• 0.1
• 0.05

• 1

• 1

• 0.15
• 0.1
• 0.05

Figure 7: β = 0.25, κNτ = 0.12, slightly disordered lattice,{α} = {(0.1, 0.1), (−0.1, 0.1), (0.1, −0.1)} at µNτ = 1.425, from left to right: scatter plots for D, history of 50 trajectories, distribution of w1.

22

Larger β and/or Nτ Same picture, weaken the transient and improve the situation:

0.1 0.2 0.3 0.4 0.5 0.6 1.35 1.4 1.45 1.5 1.55 SU3, 2 angles, beta=1.5, case 0, kappa=0.12, vs mu: ReO1/3 ex ReO1/3 ReO1/3(d) ReO-1/3 ex ReO-1/3 ReO-1/3(d)

• ReO2/3 ex
• ReO2/3
• ReO2/3(d)
• ReO-2/3 ex
• ReO-2/3
• ReO-2/3(d)

0.1 0.2 0.3 0.4 0.5 0.6 1.35 1.4 1.45 1.5 1.55 SU3, 2 angles, beta=1.5, Nt=16, case 0, kappa=0.12, vs mu: ReO1/3 ex ReO1/3 ReO1/3(d) ReO-1/3 ex ReO-1/3 ReO-1/3(d)

• ReO2/3 ex
• ReO2/3
• ReO2/3(d)
• ReO-2/3 ex
• ReO-2/3
• ReO-2/3(d)

Figure 8: β = 0.75, κNτ = 0.12, αi = 0: Nτ = 8 (left plot) and Nτ = 16 (right plot), observables vs µNτ : all trajectories (CL) and trajectories with dmin > dc (CLD).

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Expansion (regularised): The fermionic part of the drift is of the form K = ∂D D−1 + ∂ ˜ D ˜ D−1 (27) We write D = 1 + X, X = a P + b P ′, ˜ D = 1 + ˜ X, ˜ X = ˜ a P + ˜ b P ′ and center the Taylor expansion at the shifted point X = λ, ( ˜ X = ˜ λ) 1 D = 1 λ + 1

∞

• n=0

λ − X λ + 1 n , 1 ˜ D = 1 ˜ λ + 1

∞

• n=0

˜ λ − ˜ X ˜ λ + 1 n (28)

24

X

pole

2 1

λ

Figure 9: 1, 2: points in the analyticity domain of D−1 which are not reached by the usual Taylor expansion, but by the shifted one. For λ = 0 (no regularization) the expansion shows run-aways and does not converge. With λ = i(a + b) the convergence is lousy, with λ = a + b the convergence is very good, reproduces exact results.

25

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 10 12 14 16 18 20 22 24 26 \beta=0.5, \kappaL=0.12, \muL=1.425, $\lambda=i(a+b)$, vs q: O1 ex O-1 ex O2 ex O-2 ex O3 ex O-3 ex O1 O-1 O2 O-2 O3 O-3

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 10 12 14 16 18 20 22 24 26 \beta=0.5, \kappaL=0.12, \muL=1.425, $\lambda=a+b$, vs q: O1 ex O-1 ex O2 ex O-2 ex O3 ex O-3 ex O1 O-1 O2 O-2 O3 O-3

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.41 1.42 1.43 1.44 1.45 1.46 \beta=0.5, \kappaL=0.12, q=16, zreg=i(a+b), vs \muL O1 ex O-1 ex O2 ex O-2 ex O3 ex O-3 ex O1 O-1 O2 O-2 O3 O-3 O1 O-1 O2 O-2 O3 O-3

• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 1.2 1.41 1.42 1.43 1.44 1.45 1.46 \beta=0.5, \kappaL=0.12, q=16, zreg=(a+b), vs \muL: O1 ex O-1 ex O2 ex O-2 ex O3 ex O-3 ex O1 O-1 O2 O-2 O3 O-3

Figure 10: Observables from expansion, β = 0.25, κNτ = 0.12, αi = 0. Upper plots: vs expansion order q at µNτ = 1.425. Lower plots vs µNτ at q = 22. Left: λ = i(a + b) , Right: λ = (a + b).

26

Tentative conclusions

• Two regions:

Re D > 0 red fish: correct, Re D < 0 blue whiskers: transient, separated by bottle neck with small dmin Solutions:

• Long thermalization, discard blue whiskers, start in the red fish,
• Use regularised expansion centered in the red fish region,

lead to correct results.

• Similar behaviour for other parameters.
• Similar behaviour in the U(1) one-link model.
• −

→ Possible generalization, at least for HD-QCD.

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• 4. Lattice QCD with chemical potential,

HD approximation, Wilson fermions

28

Hopping parameter expansion, resummed loop expansion det W = exp  −

∞

• l=1
• {Cl}

∞

• s=1

(κlgCl)

s

s Tr

• Ls

Cl

• 

 (29) =

∞

• l=1
• {Cl}

Det D,C

• 1

I − (κ)lgClLCl

• .

(30) Here Cl are non-repeating, closed chains of links of length l, κ hopping parameter, r winding number in the ±4, ǫ = 1(−1) for pbc (apbc) and a loop LCl can cover Cl s-times before Dirac and colour traces close: LCl =

• λ∈Cl

ΓλUλ gCl =

• ǫ e±Nτ µr

Γ±µ = 1 ± γµ (31) Notice: This formula involves infinite series and the zeroes suggested by Eq. (30) are not directly zeroes of det W. Cutting the product at some order produces an approximation of det W to this order. This can be seen as a well defined model, or term in a systematic approximation.

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The lattice HD-QCD model I. Bender, T. Hashimoto, F. Karsch,

• V. Linke, A. Nakamura, M. Plewnia, IOS, W. Wetzel, 1991:

In the limit κ → 0, µ → ∞, ζ = κ eµ : fixed only Polyakov loops survive in the loop expansion and the determinant factorizes.

(κ (κ (κ κ e e e−µ µ µ ) ) ) N N N n τ τ τ

τ σ

2 κ

T = 1 / N a

τ τ

Figure 11: The HDM - QCD model: 0-th and 2-nd order. κ2 , κ4 orders: straightforward (beware combinatorics).

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HD-QCD applications

• Fixed charge (”quenched finite density QCD”), J. Engels, O.

Kaczmarek, F. Karsch, E. Laermann NPhProc 1999

• Phase diagram, quark prop., (full Y-M action, ph. RW; compare MF

and order β s.c.) R. De Pietri, A. Feo, E. Seiler, IOS, PRD 2007;

• G. Aarts and I0S, JHEP 2008)
• Effective action from β and κ expansion, reweighting and CLE, phase

diagram and further results P. de Forcrand, M. Fromm, PRL 2010;

• M. Fromm, J. Langelage, S. Lottini, O. Philipsen, JHEP 2012;
• J. Langelage, M. Neuman and O. Philipsen, 2015
• CLE using gauge cooling, full Y-M action E. Seiler, D. Sexty,

IOS, PLB 2013; B. J´ ager et al, D. Sexty, ... Further: Gluons and light quarks on heavy dense background, Upper corner of Columbia plot, Dense mattter, Neutron stars, ..

31

Notice that the HD-QCD determinant det W0(µ) is just a product of local determinants of the form in the effective model, S = β 6 SY M({U}) + ln det W0(µ) (32) det W0(µ) ≡

• x

Det (1 I + CPx)2 Det

• 1

I + ˜ CP−1

x

2 (33) therefore the results for the latter are relevant here, as illustrated in the following figures. Notice that it is important to use the resummation Eq. (30) since the coefficients C = (2κ)Nτ gP = (2κeµ)Nτ can be > 1.

32

0.2 0.4 0.6 0.8 1 1 2 3 4 5 µ 83*6 lattice β=5.9 κ=0.12 α=1 12 g.c. steps n/nsat average phase 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 2 3 4 5 µ 83*6 lattice β=5.9 κ=0.12 α=1 12 g.c. steps Polyakov loop Polyakov loop inverse strong coupling P strong coupling P’

Figure 12: HD-QCD, β = 5.9, κ = 0.12, 836 lattice: Baryon density and average phase vs. µ; P, P′ vs. µ (solid lines: analytic strong coupling result).

33

1e-05 0.0001 0.001 0.01 0.1 1 10 1e-05 0.0001 0.001 0.01 0.1 1 10 100 Histogram of the absolute value of local determinants in HDQCD 84 lattice β=6.0 κ=0.12 NF=1 µ=1.1 µ=1.3 µ=1.425 µ=1.6 x1.6

Figure 13: Local determinants in HD-QCD, 84 lattice, β = 6, κ = 0.12: histograms at various µ, scatterplot at µ = 1.425. (Figures provided by

• D. Sexty.)

At this β dmin stays rather large and the scatter plot shows no

• whiskers. The results are correct (comparison with reweighting).

Further results in the other talks from our group.

34

• 5. Discussion

35

Meromorphic drifts may but do not necessarily lead to wrong results:

• P(x, y) is found to vanish at the zeroes of ρ(z) and thus on the poles
• f the drift. If the evolution of the observables does not produce

essential singularities the proof of equivalence goes through and the results are correct. The former is, however, difficult to check.

• In an effective model we found that in a certain parameter region the

process acquires a transient with Re D < 0 producing wrong results, and that avoiding the transient leads to correct results - maybe only approximately, especially if the metastability of the transient is large.

• The behaviour seen in the effective model is relevant for HD-QCD,

possibly also for higher orders and for full QCD, especially when approaching the latter in the frame of a certain systematic expansion.

• There might be further ways of dealing with meromorphy - see

related talks at this workshop.

36

• 6. Appendix: some more results

37

−S = β

3

• i=1
• eαiei wi + e−αie−i wi

+ ln Det + ln H (34)

• 0. ordered lattice: {αi = 0}.
• 1. slightly disordered lattice:

{α} = (0.1, 0.1), (−0.1, 0.1), (0.1, −0.1).

• 2. disordered lattice:

{α} = (0.7, 0.7), (−0.7, 0.7), (0.7, −0.7).

• 3. strongly disordered lattice:

{α} = (0.2, 1.5), (−0.2, 3.1), (0.2, −0.7).

38

µNτ = 1.375 case trU trU−1 trU2 trU−2 tlang CL (0.972,0.000) (1.064,-0.001) (-0.730,-0.002) (-0.537,-0.002) 600+2800 CLD (0.972,0.000) (1.065,-0.001) (-0.730,-0.002) (-0.537,-0.002) 600+2800 CLR (0.973,0.000) (1.065,-0.000) (-0.730,-0.000) (-0.538,-0.000) 1000+5500 exact (0.972,0.000) (1.065,-0.000) (-0.730,-0.000) (-0.537,-0.000) CL 1 (0.973,-0.003) (1.065, 0.004) (-0.728,-0.010) (-0.538, 0.010) 600+2800 CLD (0.973,-0.002) (1.066, 0.003) (-0.728,-0.010) (-0.538, 0.010) 600+2800 CLR (0.974,-0.002) (1.067, 0.003) (-0.729,-0.010) (-0.538, 0.010) 1000+5500 exact (0.971,-0.001) (1.068, 0.002) (-0.736,-0.007) (-0.532, 0.006) CL 2 (0.974,-0.035) (1.034, 0.017) (-0.685,-0.071) (-0.607, 0.066) 600+2800 CLD (0.974,-0.035) (1.036, 0.014) (-0.690,-0.067) (-0.603, 0.062) 600+2800 CLR (0.978,-0.036) (1.033, 0.019) (-0.678,-0.079) (-0.615, 0.073) 1000+5500 exact (0.985,-0.069) (1.068, 0.008) (-0.720,-0.132) (-0.566, 0.093) CL 3 (0.667,-0.005) (0.830, 0.022) (-0.779,-0.036) (-0.490, 0.030) 1000+5500 CLD (0.666,-0.011) (0.828, 0.031) (-0.777,-0.035) (-0.490, 0.055) 1000+5500 CLR (0.669,-0.006) (0.831, 0.022) (-0.780,-0.039) (-0.489, 0.033) 1000+5500 exact (0.660, 0.012) (0.823, 0.014) (-0.774,-0.010) (-0.488,-0.004)

Table 2: β = 0.25 , κNτ = 0.12, µNτ = 1.375

39

µNτ = 1.425 case trU trU−1 trU2 trU−2 tlang CL (0.885,-0.009) (0.892,0.009) (-0.597,0.002) (-0.595,-0.002) 600+2800 CLD (1.069, 0.000) (1.073,0.000) (-0.648,-0.000) (-0.639,-0.002) 600+2800 CLR (1.069, 0.000) (1.072, 0.000) (-0.649,-0.001) (-0.641,-0.001) 1000+5500 CL long (1.063,-0.017) (1.067,-0.002) (-0.648, 0.002) (-0.640,-0.003) 0+11000 CLD long (1.068, 0.000) (1.072, 0.001) (-0.648,-0.000) (-0.640, 0.003) 0+11000 exact (1.069, 0.000) (1.073, 0.000) (-0.649,-0.000) (-0.640, 0.000) CL 1 (0.958,-0.006) (0.969,-0.001) (-0.627,-0.005) (-0.608,-0.005) 600+2800 CLD (1.066,-0.005) (1.075,-0.000) (-0.657,-0.001) (-0.633,-0.002) 600+2800 CLR (1.066,-0.002) (1.074,-0.001) (-0.656,-0.005) (-0.635,-0.005) 1000+5500 exact (1.067,-0.003) (1.075,-0.002) (-0.654,-0.006) (-0.635,-0.006) CL 2 (1.033,-0.005) (1.038,-0.000) (-0.647,-0.041) (-0.644,-0.027) 600+2800 CLD (1.058,-0.005) (1.062,-0.000) (-0.654, 0.040) (-0.649,-0.025) 600+2800 CLR (1.057,-0.005) (1.062,-0.019) (-0.654, 0.042) (-0.650,-0.027) 1000+5500 exact (1.079,-0.073) (1.074,-0.006) (-0.638,-0.127) (-0.667,-0.083) CL 3 (0.727,-0.039) (0.747, 0.044) (-0.666,-0.036) (-0.626, 0.030) 1000+5500 CLD (0.805,-0.010) (0.825, 0.015) (-0.693,-0.033) (-0.649, 0.032) 1000+5500 CLR (0.794,-0.011) (0.814, 0.015) (-0.690,-0.035) (-0.645, 0.034) 1000+5500 exact (0.794, 0.007) (0.809, 0.012) (-0.684,-0.007) (-0.654,-0.010)

Table 3: β = 0.25 , κNτ = 0.12, µNτ = 1.425

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β = 0.75 case trU trU−1 trU2 trU−2 tlang µNτ = 1.375 CL (1.465, 0.000) (1.498,-0.002) (-0.467,-0.002) (-0.372,-0.002) 400+2400 CLD (1.465, 0.000) (1.498,-0.002) (-0.467,-0.002) (-0.372,-0.002) 400+2400 CLR (1.464,0.000) (1.497,-0.000) (-0.470,-0.000) (-0.374,-0.001) 400+2400 exact (1.464,0.000) (1.497,-0.000) (-0.469,-0.000) (-0.373,-0.000) CL 1 (1.465,-0.008) (1.493, 0.004) (-0.458,-0.026) (-0.389, 0.020) 400+2400 CLD (1.465,-0.008) (1.493, 0.003) (-0.458,-0.026) (-0.389, 0.020) 400+2400 CLR (1.465,-0.005) (1.494, 0.005) (-0.461,-0.018) (-0.386, 0.018) 400+2400 exact (1.462,-0.007) (1.501, 0.002) (-0.478,-0.019) (-0.364, 0.009) µNτ = 1.425 CL (1.343, 0.051) (1.346,-0.001) (-0.383,-0.021) (-0.384,-0.020) 400+2400 CLD (1.510,-0.000) (1.510,-0.000) (-0.409,-0.000) (-0.405,-0.000) 400+2400 CLR (1.511,-0.000) (1.512, 0.000) (-0.407, 0.001) (-0.403,-0.001) 400+2400 exact (1.510,-0.000) (1.511,-0.000) (-0.409, 0.000) (-0.405,-0.000) CL 1 (1.359,-0.003) (1.366, 0.009) (-0.394,-0.019) (-0.385,-0.002) 400+2400 CLD (1.506,-0.007) (1.510, 0.000) (-0.416,-0.021) (-0.405,-0.002) 400+2400 CLR (1.510,-0.007) (1.512, 0.000) (-0.413,-0.021) (-0.403,-0.000) 400+2400 exact (1.507,-0.007) (1.514, 0.000) (-0.417,-0.000) (-0.405, 0.000)

Table 4: β = 0.75 , κNτ = 0.12

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β = 1.25 case trU trU−1 trU2 trU−2 tlang µNτ = 1.375 CL (1.826, 0.000) (1.838,-0.001) (-0.070,-0.000) (-0.024,-0.002) 400+2400 CLD (1.826, 0.000) (1.838,-0.001) (-0.070,-0.002) (-0.024,-0.002) 400+2400 CLR (1.826, 0.000) (1.838,-0.000) (-0.072,-0.002) (-0.026,-0.000) 400+2400 exact (1.825, 0.000) (1.837,-0.000) (-0.072,-0.000) (-0.026,-0.000) CL 1 (1.828,-0.007) (1.834, 0.003) (-0.053,-0.027) (-0.043, 0.020) 400+2400 CLD (1.828,-0.007) (1.834, 0.003) (-0.053,-0.027) (-0.043, 0.020) 400+2400 CLR (1.824,-0.009) (1.830, 0.002) (-0.062,-0.030) (-0.052, 0.019) 400+2400 exact (1.824,-0.014) (1.838,-0.005) (-0.076,-0.035) (-0.022, 0.019) µNτ = 1.425 CL (1.669,-0.010) (1.672, 0.010) (-0.028,-0.005) (-0.031,-0.005) 400+2400 CLD (1.846, 0.000) (1.846, 0.000) (-0.033, 0.050) (-0.031, 0.001) 400+2400 CLR (1.847, 0.001) (1.847,-0.001) (-0.029,-0.000) (-0.029,-0.000) 400+2400 exact (1.846, 0.000) (1.847, 0.000) (-0.033,-0.000) (-0.031, 0.000) CL 1 (1.708,-0.027) (1.712,-0.028) (-0.033,-0.000) (-0.034, 0.005) 400+2400 CLD (1.844,-0.002) (1.845, 0.004) (-0.037,-0.010) (-0.034, 0.015) 400+2400 CLR (1.845,-0.002) (1.846, 0.000) (-0.034,-0.029) (-0.027, 0.015) 400+2400 exact (1.845,-0.003) (1.848,-0.006) (-0.037,-0.034) (-0.027, 0.000)

Table 5: β = 1.25 , κNτ = 0.12

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