Computation for the lace expansion for percolation

Analysis of Section 3.3

Robert Fitzner

In this file we performs we the computations of the analysis of the non-backtracking lace expansion for percolation. All references in this version of the notebook are to the PhD thesis of the author.

We expect as input the dimension d and the constance (Subscript[Γ, 1]),(Subscript[Γ, 2]),(Subscript[Γ, 3]), (Subscript[c, 1]),..,(Subscript[c, 4]). After choosing these quantities select the menu item Evaluate-> Evaluate Notebook. When thse computations/evaluationg are finished the results are show in tables at the end of the document. There we also show whether the bootstrap and thereby the analysis is succesful.

We first compute bounds on the simple random walk two-point function (Section 5.2.1). Then we compute bound on the two-point function and repulsive diagrams (Section 5.1.2). We use these bounds to compute the Diagramatic bounds derived in Section 4.5. and compute the bounds used for the Analysis in Section 3.3.

10.5.2013

Input

The dimension in which we perfrom the computations

d=38;

For the bootstrap we assume that (Subscript[f, i])(z)≤ (Subscript[Γ, i]) with (Subscript[Γ, i]) gives as follows

Gamma1=1.0152616;Gamma2=1.023744;Gamma3=1.2;

To define the bootstrap function (Subscript[f, 3]) we use the following constants

c1=0.2256;
c2=0.6994;
c3=0.074216;
c4=9.12551;

{38, 1.01526, 1.02374, 1.2, 0.225599, 0.699387, 0.0742245, 9.12546}

Values that are working for d=38 (* deactivated*)

(*d=38;
Gamma1=1.0152616;Gamma2=1.023744;Gamma3=1.2;
c1=0.2256;
c2=0.6994;
c3=0.074216;
c4=9.12551;*)

Simple Random Walk integral

We compute the two-point function of the simple random walk,

(Subscript[I, n,m])(x)=(∫_([ -ℼ,ℼ]))(e^(ik x))((((OverHat[D])^m) (k))/(((1-(OverHat[D])(k)))^n))(((d^d)k)/(((2π))^d))

see Section 5.2, using that

(Subscript[I, n,m])(x)=(Subscript[I, n,(m-1)])(x)-(Subscript[I, (n-1),(m-1)])(x)
(Subscript[I, n,0])(x)=(1/((n-1)!))(∫_0^∞)(t^(n-1))(∏_(i=1)^d)(∫_(-π)^π)(e^(-t/d(1-cos((Subscript[k, i]))))(e^((Subscript[ιk, i])(Subscript[x, i])))(dk/(2π)) =(1/((n-1)!))(∫_0^∞)(t^(n-1))(∏_(i=1)^d)F(t,d,(Abs[(Subscript[x, i])]))

where F(t,d,n) is the modified Besselfunction. We implement the Besselfunciton and a function to compute (Subscript[I, n,0])(x).

F[t_,d_,N_]:= (E^(-(t/d))) BesselI[N,(t/d)];
NInt[n_,d_,T_]:=1/((n-1)!)*NIntegrate[t^(n-1)*(F[t,d,0])^d,{t,0,T},WorkingPrecision->40];

Then we define the number of n-step SRW loop as given in Section (5.2.6)-(5.2.10)

s2=N[2d];
s4=N[(d*((4!)/(2 2))+((d(d-1))/2)*4!)];
s6=N[ (d*((6!)/(3!3!))+d*(d-1)*((6!)/(2 2))+((d(d-1)(d-2))/(3!))*6!)];
s8=N[(d*((8!)/(4!4!))+d*(d-1)*(((8!)/(3!3!))+((8!)/(2^5)))+((d(d-1)(d-2))/2)*((8!)/(2 2))+((d(d-1)(d-2)(d-3))/(4!))*8!)];

Then we compute (Subscript[I, n,0])(0) for n=1,2,3,4:

I10=NInt[1,d,∞];
I20=NInt[2,d,∞];
I30=NInt[3,d,∞];
I40=NInt[4,d,∞];

and use (Subscript[I, n,m])(0)=(Subscript[I, n,(m-1)])(0)-(Subscript[I, (n-1),(m-1)])(0) to compute (Subscript[I, n,m])(0):

SRWTwoPointFunctionTable = {{ n m,0, 1, 2, 3, 4}, {0,1, I10, I20, I30, I40}, {1,0,0,0, 0, 0}, {2,(s2/(((2d))^2)), 0,0, 0,0}, {3,0, 0,0, 0, 0}, {4,(s4/(((2d))^4)), 0,0, 0, 0},{5, 0,0,0, 0, 0},{6,(s6/(((2d))^6)),0,0, 0, 0},{7, 0,0,0, 0, 0},{8,(s8/(((2d))^8)),0,0, 0, 0},{9, 0,0,0, 0, 0},{10,-1,0,0, 0, 0}};
For[i=3,i<13,i++,
For[j=3,j<7,j++,
SRWTwoPointFunctionTable[[i,j]]=SRWTwoPointFunctionTable[[i-1,j]]-SRWTwoPointFunctionTable[[i-1,j-1]];]
]
Clear[i,j]

I11=SRWTwoPointFunctionTable[[3,3]];
I12=SRWTwoPointFunctionTable[[4,3]];
I14=SRWTwoPointFunctionTable[[6,3]];
I16=SRWTwoPointFunctionTable[[8,3]];
I16=SRWTwoPointFunctionTable[[8,3]];
I18=SRWTwoPointFunctionTable[[10,3]];
I110=SRWTwoPointFunctionTable[[12,3]];
I21=SRWTwoPointFunctionTable[[3,4]];
I22=SRWTwoPointFunctionTable[[4,4]];
I24=SRWTwoPointFunctionTable[[6,4]];
I26=SRWTwoPointFunctionTable[[8,4]];
I28=SRWTwoPointFunctionTable[[10,4]];
I210=SRWTwoPointFunctionTable[[12,4]];
I31=SRWTwoPointFunctionTable[[3,5]];
I32=SRWTwoPointFunctionTable[[4,5]];
I33=SRWTwoPointFunctionTable[[5,5]];
I34=SRWTwoPointFunctionTable[[6,5]];
I36=SRWTwoPointFunctionTable[[8,5]];
I38=SRWTwoPointFunctionTable[[10,5]];
I310=SRWTwoPointFunctionTable[[12,5]];
I42=SRWTwoPointFunctionTable[[4,6]];
I44=SRWTwoPointFunctionTable[[6,6]];
I46=SRWTwoPointFunctionTable[[8,6]];
I48=SRWTwoPointFunctionTable[[8,6]];
I410=SRWTwoPointFunctionTable[[12,6]];

NForm[a_]:=NumberForm[N[a] ,5];
Labeled[Grid[Map[NForm,SRWTwoPointFunctionTable,{2}], Alignment -> {{Left, Center}, Baseline, {{2, 12}, {2, 6}} -> {"."}}, Frame -> True, Dividers -> {{2 -> True, -1 -> True}, {2 -> True}}, Spacings -> {1.5, {1.5, 1, {0.5}}}, ItemStyle -> {1 -> Bold, 1 -> Bold}, Background -> {Automatic, Automatic, {{2, 12}, {2, 6}} -> GrayLevel[0.9]}], Style["Value of the SRW two-point function.", Bold], Top] // Text

Bound on the two-point function and on repulsive diagrams

Definition of Constants

We define the constants for two setting s: we use s=i for bound on z=(Subscript[z, I]) and s=o for bound on zϵ ((Subscript[z, I]),(Subscript[z, c])):

Definition of key quantities

z[i]=(1/((2d -1 )));
z[o]=(Gamma1/((2d-1)));(* Upper bound on z and thereby also on (Subscript[z, c]) *)
(*bound on the two-point function*)
VarGamma1[i]=1;
VarGamma1[o]=Gamma1;
VarGamma2[i]=(((2d-2))/(2d-1)); (*(Subscript[G, z])(x)≤(Subscript[B, z])(x)≤ ((2d-2)/(2d-1))C(x)*)
VarGamma2[o]=Gamma2*((2d-2)/(2d-1)); (*(Subscript[(OverHat[G]), z])(k)≤ varGamma2 (OverHat[C])(k), follows from f2*)
VarGamma3[i]=VarGamma2[i]; (*for a bound on Delta at z=(Subscript[z, I]) we replace G with B in x-space before going to Fourier space*)
VarGamma3[o]=Gamma3; 
Varc1[o]=c1;Varc2[o]=c2;Varc3[o]=c3;Varc4[o]=c4;
Varc1[i]=0;Varc2[i]=0.5;Varc3[i]=0;Varc4[i]=4;

Further, we define variables to save the number of short NBWs, as given explain in Section 5.1.2 we can use the number if SAW only for SAW and LT:

c2ik=2; (*(Subscript[c, 2])((Subscript[e, 1])+(Subscript[e, 2]))*)
c4ik=4(2d-3)+2(2d-4);(*(Subscript[c, 4])((Subscript[e, 1])+(Subscript[e, 2]))*)
c6ik=16+84(2d-4)+36(2d-4)(2d-6)+6d c3i;(*(Subscript[c, 4])((Subscript[e, 1])+(Subscript[e, 2]))*)

c3i=(2d-2);(*(Subscript[c, 3])((Subscript[e, 1]))*)
c5i=(3(2d-2 )+4(2d-2)(2d-4) )+4d c3i;(*(Subscript[c, 5])((Subscript[e, 1]))*)
c7i=(14(2d-2 )+62(2d-2)(2d-4)+27(2d-2)(2d-4)(2d-6) )+8d c3i+4d c5i;(*(Subscript[c, 7])((Subscript[e, 1]))*)

Bounds on two-point function

We compute bounds as explained in Section 5.1.2. We begin by computing (Subscript[G, n,z])((Subscript[e, 1])) as in (5.1.22)-(5.1.24)

Do[
Bound[G7i,s]=c7i (((z[s]))^7)+(2d z[s])^9*VarGamma2[s]*I110;(* (Subscript[G, 7,z])((Subscript[e, 1]))*)
Bound[G5i,s]=c5i (((z[s]))^5)+Bound[G7i,s];(* (Subscript[G, 5,z])((Subscript[e, 1]))*)
Bound[G3i,s]=c3i (((z[s]))^3)+Bound[G5i,s];(* (Subscript[G, 3,z])((Subscript[e, 1]))*)
Bound[G1i,s]=z[s]+Bound[G3i,s];(* (Subscript[G, 1,z])((Subscript[e, 1]))*)
rho[Lower,s]=1-Bound[G1i,s];
rho[s]=1-z[i]-c3i ((z[i])^3)(1-z[i])-c5i ((z[i])^5)(((1-z[i]))^2);
,{s,{i,o}}]

Then we compute (Subscript[G, n,z]^1)((Subscript[e, 1])+(Subscript[e, 2])) and (Subscript[G, 4,z]^1)(2(Subscript[e, 2])), see (5.1.25)-(5.1.26) :

Do[
Bound[G8ik,s]=(d/(d-1))(2d z[o])^8VarGamma2[s]I110;(*(Subscript[G, 8])((Subscript[e, 1])+(Subscript[e, 2]))*)
Bound[G6ik,s]=c6ik z[s]^6+VarGamma2[s]I18;(*(Subscript[G, 6])((Subscript[e, 1])+(Subscript[e, 2]))*)
Bound[G4ik,s]=Bound[G6ik,s]+(c4ik-2(2d-3) )((z[s])^4);(*(Subscript[G, 4]^1)((Subscript[e, 1])+(Subscript[e, 2]))*)
Bound[G2ik,s]=Bound[G4ik,s]+(c2ik-1)((z[s])^2);(*(Subscript[G, 2]^1)((Subscript[e, 1])+(Subscript[e, 2]))*)

,{s,{i,o}}]

We compute the supremum of the two-point function as given in (5.1.27)-(5.1.31):

Do[
Bound[G6,s]=Max[c6ik ((z[s])^6),c7i ((z[s])^7)]+(2d z[s])^8VarGamma2[s]I18; (* Bound for (sup_x) (Subscript[G, 6])(x)=Max[(Subscript[G, 7])(e),(Subscript[G, 6])((Subscript[e, 1])+(Subscript[e, 2])),(sup_x) (Subscript[G, 8])(x)]*)
Bound[G4,s]=Max[c4ik ((z[s])^4),c5i ((z[s])^5)]+Bound[G6,s]; (* Bound for (sup_x)(Subscript[G, 4])(x)*)
Bound[G2,s]=Max[c2ik ((z[s])^2),c3i ((z[s])^3)]+Bound[G4,s]; (* Bound for (sup_x)(Subscript[G, 2])(x)*)
Bound[G1,s]=Max[Bound[G1i,s],Bound[G2,s]];(* Bound for (sup_x)(Subscript[G, 1])(x)*)
Bound[G4ii,s]=(2d+2)((z[s])^4)+Bound[G6,s];(* Bound for (sup_x)(Subscript[G, 4]^1)(2 (Subscript[e, 1]))*)
,{s,{i,o}}]

Closed repulsive diagrams

We define the bounds on the closed repulsive diagrams as desribed in Section 5.1.2 in (5.1.34)-(5.1.36). The bound does only depent on the total number of steps and the number of tw-point functions involved. It does not depent on the indivitual length of the pieces (Subscript[m, 1]),(Subscript[m, 2]),... and of the orientation of the arrows.

Do[
Bound[ClosedRepLoop,4,s]=2d z[s]Bound[G3i,s];
Bound[ClosedRepBubble,4,s]=((z[s])^4)(2d c3i)+3((z[s])^6)(2d c5i)+5((z[s])^8)(2d c7i)+ (((2d z[s]))^10)VarGamma2[s]I110+(((2d z[s]))^10)((VarGamma2[s])^2)I210;
Bound[ClosedRepBubble,3,s]=Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s];
Bound[ClosedRepTriangle,4,s]=(((4+1-4)(4+2-4))/2)((z[s])^4)(2d c3i)+(((6+1-4)(6+2-4))/2)((z[s])^6)(2d c5i)+(((8+1-4)(8+2-4))/2)((z[s])^8)(2d c7i)+ (((10-6)(9-6))/2)(((2d z[s]))^10)VarGamma2[s]I110+6 (((2d z[s]))^10)((VarGamma2[s])^2)I210+(((2d z[s]))^10)((VarGamma2[s])^3)I310;
Bound[ClosedRepSquare,4,s]=((z[s])^4)(2d c3i)+10((z[s])^6)(2d c5i)+35((z[s])^8)(2d c7i)+ 84(((2d z[s]))^10)VarGamma2[s]I110 +(((10-4)(9-4))/2)(((2d z[s]))^10)((VarGamma2[s])^2)I210+6(((2d z[s]))^10)((VarGamma2[s])^3)I310+(((2d z[s]))^10)((VarGamma2[s])^4)I410;
Bound[ClosedRepSquare,3,s]=Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s]+Bound[ClosedRepTriangle,4,s]+Bound[ClosedRepSquare,4,s];
Bound[ClosedRepSquare,2,s]=Bound[ClosedRepLoop,4,s]+2Bound[ClosedRepBubble,3,s]+Bound[ClosedRepTriangle,4,s]+Bound[ClosedRepSquare,3,s];
,{s,{i,o}}]

Open repulsive diagrams

Then we define the bound on the open repulsive diagrams as in (5.3.38):

Do[
Bound[OpenRepBubble,1,s]=Max[2c2ik ((z[s])^2)+4c4ik ((z[s])^4)+6c6ik ((z[s])^6),z[s]+3c3i ((z[s])^3)+5c5i ((z[s])^5)+7c7i ((z[s])^7)] +7(((2d z[s]))^8)VarGamma2[s]I18+(((2d z[s]))^8)((VarGamma2[s])^2)I28;
Bound[OpenRepBubble,2,s]=Max[c2ik ((z[s])^2)+3c4ik ((z[s])^4)+5c6ik ((z[s])^6),2c3i ((z[s])^3)+4c5i ((z[s])^5)+6c7i ((z[s])^7)] +6(((2d z[s]))^8)VarGamma2[s]I18+(((2d z[s]))^8)((VarGamma2[s])^2)I28;
Bound[OpenRepBubble,3,s]=Max[2c4ik ((z[s])^4)+4c6ik ((z[s])^6),c3i ((z[s])^3)+3c5i ((z[s])^5)+5c7i ((z[s])^7)] +5(((2d z[s]))^8)VarGamma2[s]I18+(((2d z[s]))^8)((VarGamma2[s])^2)I28;
Bound[OpenRepBubble,4,s]=Max[c4ik ((z[s])^4)+3c6ik ((z[s])^6),2c5i ((z[s])^5)+4c7i ((z[s])^7)] +4(((2d z[s]))^8)VarGamma2[s]I18+(((2d z[s]))^8)((VarGamma2[s])^2)I28;
Bound[OpenRepTriangle,1,s]=Max[3c2ik ((z[s])^2)+10c4ik ((z[s])^4)+21c6ik ((z[s])^6),z[s]+6c3i ((z[s])^3)+15c5i ((z[s])^5)+28c7i ((z[s])^7)] +(((8-1)(7-1))/2)(((2d z[s]))^8)VarGamma2[s]I18+7(((2d z[s]))^8)((VarGamma2[s])^2)I28+(((2d z[s]))^8)((VarGamma2[s])^3)I38;
Bound[OpenRepTriangle,2,s]=Max[c2ik ((z[s])^2)+6c4ik ((z[s])^4)+15c6ik ((z[s])^6),3c3i ((z[s])^3)+10c5i ((z[s])^5)+21c7i ((z[s])^7)] +(((8-2)(7-2))/2)(((2d z[s]))^8)VarGamma2[s]I18+6(((2d z[s]))^8)((VarGamma2[s])^2)I28+(((2d z[s]))^8)((VarGamma2[s])^3)I38;
Bound[OpenRepTriangle,3,s]=Max[3c4ik ((z[s])^4)+10c6ik ((z[s])^6),c3i ((z[s])^3)+6c5i ((z[s])^5)+15c7i ((z[s])^7)] +(((8-3)(7-3))/2)(((2d z[s]))^8)VarGamma2[s]I18+5(((2d z[s]))^8)((VarGamma2[s])^2)I28+(((2d z[s]))^8)((VarGamma2[s])^3)I38;
Bound[OpenRepTriangle,4,s]=Max[c4ik ((z[s])^4)+6c6ik ((z[s])^6),3c5i ((z[s])^5)+10c7i ((z[s])^7)] +(((8-4)(7-4))/2)(((2d z[s]))^8)VarGamma2[s]I18+5(((2d z[s]))^8)((VarGamma2[s])^2)I28+(((2d z[s]))^8)((VarGamma2[s])^3)I38;
Bound[OpenRepSquare,2,s]=Max[c2ik ((z[s])^2)+10c4ik ((z[s])^4)+35c6ik ((z[s])^6),3c3i ((z[s])^3)+20c5i ((z[s])^5)+56c7i ((z[s])^7)] +84(((2d z[s]))^8)VarGamma2[s]I18+(((8-2)(7-2))/2)(((2d z[s]))^8)((VarGamma2[s])^2)I28+6(((2d z[s]))^8)((VarGamma2[s])^3)I38+(((2d z[s]))^8)((VarGamma2[s])^4)I48;
,{s,{i,o}}]

Weighted Diagrams

We define weighted diagams as explained in Section 5.1., e.g. (5.1.19) and (5.1.42)-(5.1.49) we derive weighted closed diagrams

Do[
Bound[WeightedClosedBubble,4,s]=(((2d z[s]))^4)VarGamma2[s]VarGamma3[s](Varc1[s]I24+(2Varc2[s]+Varc3[s])I34);
Bound[WeightedClosedBubble,3,s]=Bound[WeightedClosedBubble,4,s]+2d ((z[s])^3)( Bound[G3i,s]+3 (2d-2)(d/(d-1))(d/(d-2))(2d z[o])^3VarGamma2[s]I16+ 5(2d-2)(d/((d-1)))Bound[G4,s]+ 9(((z[s])^3)+3(2d-2)((z[s])^5)+Bound[G6,s]));

Bound[WeightedClosedBubble,2,s]=Bound[WeightedClosedBubble,3,s]+8d ((z[s])^2)Bound[G4ii,s]+8d ((z[s])^2)(2d-2)(((z[s])^2)+Bound[G4ik,s]);
Bound[WeightedClosedBubble,0,s]=Bound[WeightedClosedBubble,2,s]+2d z[s] Bound[G3i,s];


Bound[WeightedOpenLine,0,s]=VarGamma3[s](Varc1[s]I10+(2Varc2[s]+Varc4[s])I20);
Bound[WeightedOpenBubble,0,s]=VarGamma2[s]VarGamma3[s](Varc1[s]I20+(2Varc2[s]+Varc4[s])I30);
Bound[WeightedOpenBubble,1,s]=(2d z[s])VarGamma2[s]VarGamma3[s](Varc1[s](Sqrt[I20 I22])+(2Varc2[s]+Varc4[s])(Sqrt[I30 I32]));
Bound[WeightedOpenBubble,2,s]=(((2d z[s]))^2)VarGamma2[s]VarGamma3[s](Varc1[s]I22+(2Varc2[s]+Varc4[s])I32);
Bound[WeightedOpenBubble,3,s]=(((2d z[s]))^3)VarGamma2[s]VarGamma3[s](Varc1[s](Sqrt[I22 I24])+(2Varc2[s]+Varc4[s])(Sqrt[I32 I34]));
,{s,{i,o}}]

Building Blocks

Definition of diagrams without weight

Here we define the quantities of Section 4.5.2. The element of the bounds (4.5.30)-(4.5.38). We defined the element in the thesis using bubble, triangles, square and even pentragram. The bound on these diagrams depend only on the number of two-point funtions/pieces without fix lengh and the number of fixed steps.

We define the bound on (P^((0),b))define in Table 4.18 as follows:

Do[
Bound[P,0,s]=1+Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s];
Bound[P,1,s]=2Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s];
Bound[P,2,s]=Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s]+Bound[ClosedRepTriangle,4,s];
,{s,{i,o}}];

The bound on (A^(a,b))define in Table 4.19 we declare the variables:

Do[
Bound[A,0,0,s]=Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s];
Bound[A,0,1,s]=Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s];
Bound[A,0,2,s]=Bound[ClosedRepTriangle,4,s];
Bound[A,1,0,s]=(1/(2d z[s]))(Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s]);
Bound[A,1,1,s]=(1/(2d z[s]))(Bound[ClosedRepLoop,4,s]+Bound[ClosedRepBubble,4,s]);
Bound[A,1,2,s]=((Bound[ClosedRepTriangle,4,s])/(2d z[s]));
Bound[A,2,0,s]=Bound[OpenRepBubble,1,s];
Bound[A,2,1,s]=Bound[OpenRepBubble,2,s];
Bound[A,2,2,s]=Bound[OpenRepTriangle,3,s];
,{s,{i,o}}];

We defined (A^(i,a,b)) using diagrams what do not need to be repulsive. Follow Table 4.20 we define

Do[
Bound[Ai,0,0,s]=(((2d z[s]))^4)VarGamma2[s] I14+(((2d z[s]))^4)((VarGamma2[s])^2) I24;
Bound[Ai,0,1,s]=(((2d z[s]))^4)VarGamma2[s] I14+(((2d z[s]))^4)((VarGamma2[s])^2) I24;
Bound[Ai,0,2,s]=(((2d z[s]))^4)((VarGamma2[s])^3) I34;
Bound[Ai,1,0,s]=(((((2d z[s]))^4)VarGamma2[s] I14+(((2d z[s]))^4)((VarGamma2[s])^2) I24)/(2d z[s]));
Bound[Ai,1,1,s]=(((((2d z[s]))^4)VarGamma2[s] I14+(((2d z[s]))^4)((VarGamma2[s])^2) I24)/(2d z[s]));
Bound[Ai,1,2,s]=(((2d z[s]))^3)((VarGamma2[s])^3) I34;
Bound[Ai,2,0,s]=(((2d z[s]))^2)((VarGamma2[s])^2) I22;
Bound[Ai,2,1,s]=(((2d z[s]))^2)((VarGamma2[s])^2) I22;
Bound[Ai,2,2,s]=(((2d z[s]))^3)((VarGamma2[s])^3) (Sqrt[I32 I34]);
,{s,{i,o}}];

Then we define ((OverBar[A])^(a,b))as in (4.5.16)-(4.5.20):

Do[
Do[
Bound[Abar,a,0,s]=Bound[Ai,a,0,s];
Bound[Abar,a,1,s]=((Bound[Ai,a,1,s])/(2d z[i]));
,{a,{0,1,2}}];
Bound[Abar,0,2,s]=(((2d z[s]))^2)((VarGamma2[s])^2) I22;
Bound[Abar,1,2,s]=(2d z[s])((VarGamma2[s])^2) I22;
Bound[Abar,2,2,s]=((VarGamma2[s])^2) I20;
,{s,{i,o}}];

Then we define (B^(2,i,a,b)) as in Table 4.21 and the comment after (4.5.20)

Do[
Do[
Bound[B2i,a,0,s]=0;
Bound[B2i,a,1,s]=0;
,{a,{0,1,2}}];
Bound[B2i,0,2,s]=Bound[ClosedRepBubble,3,s]((Bound[ClosedRepTriangle,4,s])/(2d z[s]))+Bound[ClosedRepTriangle,4,s]Bound[OpenRepTriangle,3,s];
Bound[B2i,1,2,s]=((Bound[ClosedRepBubble,3,s])/(2d z[s]))((Bound[ClosedRepTriangle,4,s])/(2d z[s]))+Bound[ClosedRepTriangle,4,s]((Bound[OpenRepTriangle,3,s])/(2d z[s]));
Bound[B2i,2,2,s]=((Bound[ClosedRepBubble,3,s])/(2d z[s]))Bound[OpenRepTriangle,3,s]+Bound[OpenRepBubble,2,s]Bound[OpenRepTriangle,4,s];
,{s,{i,o}}];

Further, we define ((OverBar[B])^(2,i,a,b)) as in Tables 4.22-4.23 and the comment after (4.5.20)

Do[
Do[
Bound[Bbar2i,0,a,s]=0;
,{a,{0,1,2}}];
Bound[Bbar2i,1,0,s]=Bound[G3i,s] Bound[ClosedRepBubble,4,s]+Bound[G2,s]Bound[ClosedRepTriangle,4,s];
Bound[Bbar2i,1,1,s]=Bound[G2,s]((Bound[ClosedRepTriangle,4,s])/(2d z[s]));
Bound[Bbar2i,1,2,s]=Bound[G3i,s] Bound[OpenRepBubble,3,s]+Bound[G2,s]Bound[OpenRepTriangle,4,s];
Bound[Bbar2i,2,0,s]=Bound[ClosedRepBubble,3,s]Bound[OpenRepBubble,4,s]+Bound[ClosedRepBubble,3,s]Bound[ClosedRepTriangle,4,s]+Bound[ClosedRepTriangle,4,s]Bound[OpenRepTriangle,3,s];
Bound[Bbar2i,2,1,s]=((Bound[Bbar2i,2,0,s])/(2d z[s]));
Bound[Bbar2i,2,2,s]=Bound[G1,o]Bound[OpenRepBubble,2,s]Bound[ClosedRepBubble,3,s]+Bound[OpenRepTriangle,3,s]((Bound[ClosedRepBubble,3,s])/(2d z[s]))+Bound[G1,o] Bound[OpenRepBubble,1,s]Bound[ClosedRepTriangle,4,s];
,{s,{i,o}}];

Bound[OpenRepBubble,1,o]
Bound[G1,o]

We define the bound on (P^((N),i,b))define in Table 4.24 as follows:

Do[
Bound[Piota,0,s]=2d Bound[G3i,s]Bound[P,0,s];
Bound[Piota,1,s]=2d Bound[G3i,s]Bound[P,1,s]+2d(z[s]+Bound[G3i,s]);
Bound[Piota,2,s]=2d Bound[G3i,s]Bound[P,2,s]+Bound[G3i,s]+((Bound[ClosedRepBubble,3,s])/( z[s]))+2d(z[s]+Bound[G3i,s]+((Bound[ClosedRepBubble,3,s])/(2d z[s])))Bound[ClosedRepBubble,3,s];
,{s,{i,o}}];

Definition of diagrams with weight

Now we first implement the bound in the diagrams (H^(1,a,b)), (H^(2,i,a,b)) and (H^(3,i,a,b)) defined in (4.5.23)-(4.5.25):

Do[
Bound[H1,0,0,s]=Bound[WeightedClosedBubble,0,s];
Bound[H1,1,0,s]=((Bound[WeightedClosedBubble,0,s])/(2d z[s]));
Bound[H1,0,1,s]=((Bound[WeightedClosedBubble,0,s])/(2d z[s]));
Bound[H1,1,1,s]=((Bound[WeightedClosedBubble,2,s])/(((2d z[s]))^2));
Bound[H1,2,0,s]=Bound[WeightedOpenBubble,0,s];
Bound[H1,0,2,s]=Bound[WeightedOpenBubble,0,s];

Bound[H1,2,1,s]=((Bound[WeightedOpenBubble,1,s])/(2d z[s]));
Bound[H1,1,2,s]=((Bound[WeightedOpenBubble,1,s])/(2d z[s]));
Bound[H1,2,2,s]=Bound[WeightedOpenBubble,0,s];

Do[Do[
Bound[H2,a,b,s]=Bound[H1,a,b,s];
,{a,{0,1,2}}],{b,{0,1,2}}];
(* for H3 we know that the unweighted path has at least length one*)
Bound[H3,0,0,s]=Bound[WeightedClosedBubble,0,s];
Bound[H3,1,0,s]=((Bound[WeightedClosedBubble,2,s])/(2d z[s]));
Bound[H3,0,1,s]=((Bound[WeightedClosedBubble,2,s])/(2d z[s]));
Bound[H3,1,1,s]=((Bound[WeightedClosedBubble,3,s])/(((2d z[s]))^2));
(* to use the information that the unweighted path has at least length one we use Chauchy schwarz to obtain*)
Bound[H3,2,0,s]=Bound[WeightedOpenBubble,1,s];
Bound[H3,0,2,s]=Bound[WeightedOpenBubble,1,s];

Bound[H3,1,2,s]=((Bound[WeightedOpenBubble,2,s])/(2d z[o]));
Bound[H3,2,1,s]=((Bound[WeightedOpenBubble,2,s])/(2d z[o]));
Bound[H3,2,2,s]=Bound[WeightedOpenBubble,1,s];

Clear[a,b,t];
,{s,{i,o}}]

As explained in Section 4.5.6 we bound (C^(1,i,a,b)) and (C^(2,i,a,b)) in terms of other diagram. At this point we only implement the bound on (C^(3,i,a,b)), see (4.5.90)-(4.5.94). For a=2,b=2 we implement the bound as describes in (4.5.97). For other a,b we improve this bound using three informations: 1.) By symmetrie we can we bound the contribution (C^(3,i,a,b)) in the same way as (C^(3,i,b,a)). 2.) if a or b are 1 then we can use symmetrie to create an extra (OverHat[D])(k) . 3.) the complete square consists of at least four steps to improve the bounds. Using these three points to obtain the bounds:

Do[

Bound[C3,2,2,s]=Bound[WeightedOpenLine,0,s](((2d z[s]))^2)((VarGamma2[s])^3) I32 Bound[OpenRepSquare,2,s];
Bound[C3,1,2,s]=Bound[WeightedOpenLine,0,s](((2d z[s]))^2)((VarGamma2[s])^3) I32 ((Bound[ClosedRepSquare,3,s])/(2d z[s]));
Bound[C3,2,1,s]=Bound[C3,1,2,s];
Bound[C3,1,1,s]=Bound[WeightedOpenLine,0,s]Bound[OpenRepTriangle,3,s]((Bound[ClosedRepSquare,3,s])/(2d z[s]));
Bound[C3,0,1,s]=Bound[WeightedOpenLine,0,s]Bound[OpenRepTriangle,2,s]((Bound[ClosedRepSquare,3,s])/(2d z[s]));
Bound[C3,1,0,s]=Bound[C3,0,1,s];
Bound[C3,0,0,s]=Bound[WeightedOpenLine,0,s]Bound[OpenRepTriangle,2,s] Bound[ClosedRepSquare,2,s];
Bound[C3,0,2,s]=Bound[WeightedOpenLine,0,s](((2d z[s]))^2)((VarGamma2[s])^3) I32 Bound[ClosedRepSquare,2,s];
Bound[C3,2,0,s]=Bound[C3,0,2,s];
,{s,{i,o}}]

Definition of diagrams specific for initial step iota

As next we bound the term defined in (4.5.28) and bound in (4.5.100)-(4.5.102). We improve the bound stated of the first diagram of Figure 4.29, by noting that  y=0 is not possible, so that the unweighted lines has always at least 2 steps.

Do[

Bound[hi,part1,0,s]=Bound[WeightedClosedBubble,2,s];
Bound[hi,part1,1,s]=((Bound[WeightedClosedBubble,2,s])/(2d z[s]));
Bound[hi,part1,2,s]=Bound[WeightedOpenBubble,1,s];
Do[
Bound[hi,part2,b,s]=2d  Bound[G3i,s] Bound[H2,0,b,s]+4d  Bound[G3i,s](Sum[Bound[P,a,s]  Bound[H2,a,b,s],{a,0,2}]- Bound[H2,0,b,s])+4d  Bound[G3i,s]Sum[ Bound[H1,0,a,s]Bound[Ai,a,b,s],{a,0,2}]; 
Bound[hi,part3,b,s]=((Bound[ClosedRepLoop,4,s])/(2d z[s]))Bound[H2,1,b,s]+((Bound[ClosedRepBubble,3,s])/(2d z[s]))Bound[H2,2,b,s]+2(((Bound[ClosedRepLoop,4,s])/(2d z[s]))+((Bound[ClosedRepBubble,3,s])/(2d z[s])))(Bound[Ai,0,0,s]Max[Bound[H2,1,b,s],Bound[H2,2,b,s]]+Bound[H1,0,0,s]Max[Bound[Ai,1,b,s],Bound[Ai,2,b,s]]);

Bound[hi,b,s]=Sum[Bound[hi,t,b,s],{t,{part1,part2,part3}}]
,{b,0,2}];
,{s,{i,o}}]

Then we define the bound for (h^(i,II,b)) defined in (4.5.29) and bounded in (4.5.99).

Do[
Do[
Bound[hII,b,s]= Bound[hi,b,s]+2 Sum[Bound[hi,a,s]Bound[Ai,a,b,s]+ Sum[Bound[Piota,a,s]Bound[Ai,a,c,s]Bound[H2,c,b,s],{c,0,2}],{a,0,2}]
,{b,0,2}];
,{s,{i,o}}]

Definition of vectors and matrices

Then we define the matricies we use the compute/state the bounds. First we define the matricies for which we have already computed the entries.

Do[
Vector[P,s]=Table[Bound[P,a,s],{a,{0,1,2}}];
Vector[PNT,s]=Vector[P,s]-{1,0,0};
Vector[Piota,s]=Table[Bound[Piota,a,s],{a,{0,1,2}}];

Vector[h,s]=Table[Bound[H1,0,b,s],{b,{0,1,2}}];
Vector[hi,s]=Table[Bound[hi,b,s],{b,{0,1,2}}];
Vector[hII,s]=Table[Bound[hII,b,s],{b,{0,1,2}}];

Matrix[A,s]=Table[Bound[A,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];
Matrix[Ai,s]=Table[Bound[Ai,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];
Matrix[Abar,s]=Table[Bound[Abar,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];
Matrix[B2i,s]=Table[Bound[B2i,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];
Matrix[Bbar2i,s]=Table[Bound[Bbar2i,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];

Matrix[C3,s]=Table[Bound[C3,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];
Matrix[H1,s]=Table[Bound[H1,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];
Matrix[H2,s]=Table[Bound[H2,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];
Matrix[H3,s]=Table[Bound[H3,a,b,s],{a,{0,1,2}},{b,{0,1,2}}];
,{s,{i,o}}]

Then we define the bound for B and (OverBar[B]), see (4.5.21)-(4.5.22)

Do[
Matrix[B,s]=Matrix[Ai,s].Matrix[A,s]+Matrix[Ai,s]*Bound[P,0,s]+Matrix[B2i,s];
Matrix[Bbar,s]=Matrix[Ai,s].Matrix[A,s]+Matrix[Ai,s]+Matrix[Bbar2i,s];
,{s,{i,o}}]
Clear[a,b]

Further, we define the bound on (C^1) and (C^2) as stated in (4.5.90) and (4.5.92)

Do[
Matrix[C1,s]=(2Matrix[H2,s].Matrix[A,s]+2Matrix[Ai,s].Matrix[H1,s]+Matrix[H2,s]).Matrix[Ai,s]+Matrix[H2,s].Matrix[Bbar2i,s];
Matrix[C2,s]=(2Matrix[H3,s].Matrix[A,s]+2Matrix[Ai,s].Matrix[H1,s]+Matrix[H3,s]).Matrix[Ai,s]+2 Matrix[C3,s] + Matrix[H3,s].Matrix[Bbar2i,s];
,{s,{i,o}}]
Clear[a,b]

We compute the eigensystem of the maticies B and (OverBar[B]) to sum the bounds as explained in Section 5.3.

Do[
EigensystemB[s] = Eigensystem[Transpose[Matrix[B,s]]];
EigensystemBbar[s] = Eigensystem[Matrix[Bbar,s]];
InverseProductB[s]=Inverse[Transpose[EigensystemB[s][[2]]]].Vector[P,s];
InverseProductBForPIota[s]=Inverse[Transpose[EigensystemB[s][[2]]]].Vector[Piota,s];
InverseProductBbar[s]=Inverse[Transpose[EigensystemBbar[s][[2]]]].Vector[P,s];

Do[
EigenVectorB[j,s]=EigensystemB[s][[2,j]]*InverseProductB[s][[j]];
EigenVectorBbar[j,s]=EigensystemBbar[s][[2,j]]*InverseProductBbar[s][[j]];
EigenVectorBForPIota[j,s]=EigensystemB[s][[2,j]]*InverseProductBForPIota[s][[j]];

EigenValueB[j,s]=EigensystemB[s][[1,j]];
EigenValueBbar[j,s]=EigensystemBbar[s][[1,j]];
EigenValueBForPIota[j,s]=EigensystemB[s][[1,j]];
,{j,1,3}];
,{s,{i,o}}]
Clear[a,b]

Bound on the coefficients

Bound for k=0

Then we implement the bounds stated in Lemma 4.5.1.-4.5.2 and Proposition 4.5.3. explixitely treat N=2,3 as they will also recieve special attentention when we compute the bound on Ξ(0)-Ξ(k)

Do[
Bound[Xi,normal, 0, s]=Bound[P,0,s]-1; (* For the analysis we explicitly extract the contribution of (Subscript[δ, 0,x])*)
Bound[Xi,iota, 0, s]=Bound[G3i,s]Bound[P,0,s]; 

Bound[Xi,normal, 1, s]=Vector[P,s].Matrix[Abar,s].Vector[P,s];
Bound[Xi,iota, 1, s]=(1/(2d))Vector[Piota,s].Matrix[Abar,s].Vector[P,s];

Bound[Xi,normal, 2, s]=Vector[P,s].Matrix[B,s].Matrix[Abar,s].Vector[P,s];
Bound[Xi,iota, 2, s] =(1/(2d))Vector[Piota,s].Matrix[B,s].Matrix[Abar,s].Vector[P,s];

Bound[Xi,normal, 3, s]=Vector[P,s].Matrix[B,s].Matrix[B,s].Matrix[Abar,s].Vector[P,s];
Bound[Xi,iota, 3, s] =(1/(2d))Vector[Piota,s].Matrix[B,s].Matrix[B,s].Matrix[Abar,s].Vector[P,s];

,{s,{i,o}}]

We compue the sum over all odd and even N=> 4

Do[
Bound[Xi,normal, EvenTail, s]= Sum[(( ((EigenValueB[j,s])^3))/(1- ((EigenValueB[j,s])^2))) EigenVectorB[j,s].Matrix[Abar,s].Vector[P,s],{j,{1,2,3}}];
Bound[Xi,normal, OddTail, s]= Sum[(( ((EigenValueB[j,s])^4))/(1- ((EigenValueB[j,s])^2))) EigenVectorB[j,s].Matrix[Abar,s].Vector[P,s],{j,{1,2,3}}];
Bound[Xi,iota, EvenTail, s] =(1/(2d))Sum[(( ((EigenValueBForPIota[j,s])^3))/(1- ((EigenValueBForPIota[j,s])^2))) EigenVectorBForPIota[j,s].Matrix[Abar,s].Vector[P,s],{j,{1,2,3}}];
Bound[Xi,iota,  OddTail, s] =(1/(2d))Sum[((  ((EigenValueBForPIota[j,s])^4))/(1-  ((EigenValueBForPIota[j,s])^2))) EigenVectorBForPIota[j,s].Matrix[Abar,s].Vector[P,s],{j,{1,2,3}}];
,{s,{i,o}}]

Then we compute the sum over all odd/even N

Do[
Do[
Bound[Xi,a, Even, s]=Sum[Bound[Xi,a,t, s],{t,{0,2,EvenTail}}];
Bound[Xi,a, Odd, s]=Sum[Bound[Xi,a,t, s],{t,{1,3,OddTail}}];
Bound[Xi,a, Absolut, s]=Sum[Bound[Xi,a,t, s],{t,{Odd,Even}}];
(*Print[Bound[Xi,a, Absolut, s]];*)
,{a,{normal,iota}}]
,{s,{i,o}}]

Bound for 0 - k

Bounds for (OverHat[Ξ])(0)-(OverHat[Ξ])(k)

First we compute the terms for N=0,1,2,3 and extract therefore the contributions with trivial first and last triangle

Do[
Bound[Xi,normal,0,Delta,0,s]=Bound[WeightedClosedBubble,0,s];
Bound[Xi,normal,1,Delta,0,s]=Bound[H2,0,0,s]+4 Sum[Bound[H3,0,b,s]Bound[Ai,0,b,s],{b,0,2}]+6Vector[h,s].Matrix[Ai,s].Vector[PNT,s]+3Vector[PNT,s].Matrix[H2,s].Vector[PNT,s];(*for this bound we use the spatial symmetrie of the diagrams, when one triangle it trivial to reduce the factor 8 stated in (4.5.47) to 4 (sinis terms cancel, so we can split the cosin without creating the factor 2)*)
Bound[Xi,normal,2,Delta,0,s]=4(Vector[h,s].Matrix[Bbar,s].Matrix[Ai,s].Vector[PNT,s]+Vector[PNT,s].Matrix[B,s].(Matrix[H3,s].Vector[PNT,s]+Matrix[Ai,s].Vector[h,s])+Vector[PNT,s].Matrix[C1,s].Vector[PNT,s])+6(Vector[h,s].Matrix[Bbar,s].Matrix[Ai,s].{1,0,0}+Vector[PNT,s].Matrix[B,s].Matrix[H2,s].{1,0,0}+Vector[PNT,s].Matrix[C1,s].{1,0,0})+2({1,0,0}.Matrix[B,s].Matrix[H2,s].{1,0,0}+{1,0,0}.Matrix[C1,s].{1,0,0});
Bound[Xi,normal,3,Delta,0,s]=5(Vector[h,s].Matrix[Bbar,s].Matrix[Bbar,s].Matrix[Ai,s].Vector[PNT,s]+Vector[PNT,s].Matrix[B,s].Matrix[B,s].(Matrix[H2,s].Vector[PNT,s]+Matrix[Ai,s].Vector[h,s])+Vector[PNT,s].(Matrix[C1,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C2,s]).Vector[PNT,s])+4({1,0,0}.Matrix[B,s].Matrix[B,s].(Matrix[H2,s].Vector[PNT,s]+Matrix[Ai,s].Vector[h,s])+{1,0,0}.(Matrix[C1,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C2,s]).Vector[PNT,s])+4(Vector[h,s].Matrix[Bbar,s].Matrix[Bbar,s].Matrix[Ai,s].{1,0,0}+Vector[PNT,s].Matrix[B,s].Matrix[B,s].(Matrix[H2,s].{1,0,0})+Vector[PNT,s].(Matrix[C1,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C2,s]).{1,0,0})+3({1,0,0}.Matrix[B,s].Matrix[B,s].Matrix[H2,s].{1,0,0}+{1,0,0}.(Matrix[C1,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C2,s]).{1,0,0});
,{s,{i,o}}]

Then we compute the sum over the remaining N

Do[
Do[
v[j]=EigenVectorB[j,s];
vb[j]=EigenVectorBbar[j,s];
e[j]=EigenValueB[j,s];
eb[j]=EigenValueBbar[j,s];
,{j,{1,2,3}}];
Bound[Xi,normal,EvenTail,Delta,0,s]=Vector[h,s].Matrix[Ai,s].Sum[vb[j]*((eb[j])^3)((2/(((1-((eb[j])^2)))^2))+(4/((1-((eb[j])^2))))),{j,{1,2,3}}]+Sum[((e[j])^3)((2/(((1-((e[j])^2)))^2))+(4/((1-((e[j])^2)))))*v[j],{j,{1,2,3}}].(Matrix[H3,s].Vector[P,s]+Matrix[Ai,s].Vector[h,s])+2Sum[v[j](1/(1-((e[j])^2))),{j,{1,2,3}}].Matrix[C1,s].Sum[vb[j](1/(((1-((eb[j])^2)))^2)),{j,{1,2,3}}]+2Sum[v[j](1/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].Matrix[C1,s].Sum[vb[j](1/(1-((eb[j])^2))),{j,{1,2,3}}]-4Vector[P,s].Matrix[C1,s].Vector[P,s]+2Sum[v[j]((e[j])/(1-((e[j])^2))),{j,{1,2,3}}].Matrix[C2,s].Sum[vb[j]eb[j]((1/(((1-((eb[j])^2)))^2))+(1/((1-((eb[j])^2))))),{j,{1,2,3}}]+2Sum[v[j]((e[j])/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].Matrix[C1,s].Sum[vb[j]eb[j](1/((1-((eb[j])^2)))),{j,{1,2,3}}];

Bound[Xi,normal,OddTail,Delta,0,s]=Vector[h,s].Matrix[Ai,s].Sum[vb[j]*((eb[j])^4)((2/(((1-((eb[j])^2)))^2))+(5/((1-((eb[j])^2))))),{j,{1,2,3}}]+Sum[((e[j])^4)((2/(((1-((e[j])^2)))^2))+(5/((1-((e[j])^2)))))*v[j],{j,{1,2,3}}].(Matrix[H3,s].Vector[P,s]+Matrix[Ai,s].Vector[h,s])+Sum[v[j](2/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].(Matrix[C2,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C1,s]).Sum[vb[j](1/(1-((eb[j])^2))),{j,{1,2,3}}]+Sum[v[j](1/((1-((e[j])^2)))),{j,{1,2,3}}].(Matrix[C2,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C1,s]).Sum[vb[j]((2/(((1-((eb[j])^2)))^2))+(1/(1-((eb[j])^2)))),{j,{1,2,3}}]+
5Vector[P,s].(Matrix[C2,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C1,s]).Vector[P,s];

,{s,{i,o}}]

Bounds for (∑_i)((OverHat[Ξ])^i)(0)-((OverHat[Ξ])^i)(k)

First we compute the terms for N=0,1,2,3 and therefore extract the contributions with trivial first and last triangle

Do[
Bound[Xi,iota,0,Delta,ei,s]= 2d (2d-1)z[s]((Bound[G3i,s])^2)+(2d-1)Bound[G3i,s]Bound[WeightedClosedBubble,2,s];
Bound[Xi,iota,0,Delta,0,s]=Bound[Xi,iota,0,Delta,ei,s]+2d Bound[Xi,iota, 0, s]; (* we can use symmetrie to cancel the sin terms an obtain this bound*)

Bound[Xi,iota,1,Delta,ei,s]=Bound[hi,0,s]+Vector[hi,s].Vector[PNT,s]+Vector[Piota,s].Matrix[Ai,s].Vector[h,s];
Bound[Xi,iota,1,Delta,0,s]=Bound[Xi,iota,1,Delta,ei,s]+2d Bound[Xi,iota, 1, s];

Bound[Xi,iota,2,Delta,ei,s]=2(Vector[hII,s].Matrix[Bbar,s].{1,0,0}+Vector[Piota,s].Matrix[B,s].Matrix[H3,s].{1,0,0})+3(Vector[hII,s].Matrix[Bbar,s].Vector[PNT,s]+Vector[Piota,s].Matrix[B,s].(Matrix[H3,s].Vector[PNT,s]+Matrix[Ai,s].Vector[h,s]));
Bound[Xi,iota,2,Delta,0,s]=3(Vector[hII,s].Matrix[Bbar,s].{1,0,0}+Vector[Piota,s].Matrix[B,s].Matrix[H3,s].{1,0,0})+4(Vector[hII,s].Matrix[Bbar,s].Vector[PNT,s]+Vector[Piota,s].Matrix[B,s].(Matrix[H3,s].Vector[PNT,s]+Matrix[Ai,s].Vector[h,s]))+3 Vector[Piota,s].Matrix[B,s].Matrix[Abar,s].{1,0,0}+4 Vector[Piota,s].Matrix[B,s].Matrix[Abar,s].Vector[PNT,s] ;

Bound[Xi,iota,3,Delta,ei,s]=3(Vector[hII,s].Matrix[Bbar,s].Matrix[Bbar,s].{1,0,0}+Vector[Piota,s].Matrix[B,s].Matrix[B,s].Matrix[H3,s].{1,0,0})+3 Vector[Piota,s].Matrix[B,s].Matrix[C2,s].{1,0,0}+4(Vector[hII,s].Matrix[Bbar,s].Matrix[Bbar,s].Vector[PNT,s]+Vector[Piota,s].Matrix[B,s].Matrix[B,s].(Matrix[H3,s].Vector[PNT,s]+Matrix[Ai,s].Vector[h,s]))+4 Vector[Piota,s].Matrix[B,s].Matrix[C2,s].Vector[PNT,s];
Bound[Xi,iota,3,Delta,0,s]=4(Vector[hII,s].Matrix[Bbar,s].Matrix[Bbar,s].{1,0,0}+Vector[Piota,s].Matrix[B,s].Matrix[B,s].Matrix[H3,s].{1,0,0})+4 Vector[Piota,s].Matrix[B,s].Matrix[C2,s].{1,0,0}+4 Vector[Piota,s].Matrix[B,s].Matrix[B,s].Matrix[Abar,s].{1,0,0}+5(Vector[hII,s].Matrix[Bbar,s].Matrix[Bbar,s].Vector[PNT,s]+Vector[Piota,s].Matrix[B,s].Matrix[B,s].(Matrix[H3,s].Vector[PNT,s]+Matrix[Ai,s].Vector[h,s]))+5 Vector[Piota,s].Matrix[B,s].Matrix[C2,s].Vector[PNT,s]+5 Vector[Piota,s].Matrix[B,s].Matrix[B,s].Matrix[Abar,s].Vector[PNT,s];
,{s,{i,o}}]

Do[
Do[
v[j]=EigenVectorBForPIota[j,s];
vb[j]=EigenVectorBbar[j,s];
e[j]=EigenValueBForPIota[j,s];
eb[j]=EigenValueBbar[j,s];
,{j,{1,2,3}}];
Bound[Xi,iota,EvenTail,Delta,0,s]=Vector[hII,s].Sum[vb[j]*((eb[j])^3)((2/(((1-((eb[j])^2)))^2))+(3/((1-((eb[j])^2))))),{j,{1,2,3}}]+Sum[((e[j])^3)((2/(((1-((e[j])^2)))^2))+(3/((1-((e[j])^2)))))*v[j],{j,{1,2,3}}].(Matrix[H3,s].Vector[P,s]+Matrix[Ai,s].Vector[h,s])+Sum[v[j]((2 e[j])/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].(Matrix[C2,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C1,s]).Sum[vb[j](1/(1-((eb[j])^2))),{j,{1,2,3}}]+Sum[v[j]((e[j])/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].(Matrix[C2,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C1,s]).Sum[vb[j]((2/(((1-((eb[j])^2)))^2))+(1/(1-((eb[j])^2)))),{j,{1,2,3}}];
Bound[Xi,iota,EvenTail,Delta,ei,s]=Vector[hII,s].Sum[vb[j]*((eb[j])^3)((2/(((1-((eb[j])^2)))^2))+(4/((1-((eb[j])^2))))),{j,{1,2,3}}]+Sum[((e[j])^3)((2/(((1-((e[j])^2)))^2))+(4/((1-((e[j])^2)))))*v[j],{j,{1,2,3}}].(Matrix[H3,s].Vector[P,s]+Matrix[Ai,s].Vector[h,s])+Sum[v[j]((2 e[j])/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].(Matrix[C2,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C1,s]).Sum[vb[j](1/(1-((eb[j])^2))),{j,{1,2,3}}]+Sum[v[j]((e[j])/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].(Matrix[C2,s].Matrix[Bbar,s]+Matrix[B,s].Matrix[C1,s]).Sum[vb[j]((2/(((1-((eb[j])^2)))^2))+(2/(1-((eb[j])^2)))),{j,{1,2,3}}]+Sum[((e[j])^3)((2/(((1-((e[j])^2)))^2))+(4/((1-((e[j])^2)))))*v[j],{j,{1,2,3}}].Matrix[Ai,s].Vector[P,s];

Bound[Xi,iota,OddTail,Delta,0,s]=Vector[hII,s].Sum[vb[j]*((eb[j])^4)((2/(((1-((eb[j])^2)))^2))+(4/((1-((eb[j])^2))))),{j,{1,2,3}}]+Sum[((e[j])^4)((2/(((1-((e[j])^2)))^2))+(4/((1-((e[j])^2)))))*v[j],{j,{1,2,3}}].(Matrix[H3,s].Vector[P,s]+Matrix[Ai,s].Vector[h,s])+Sum[v[j]((2 e[j])/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].Matrix[C2,s].Sum[vb[j](1/(1-((eb[j])^2))),{j,{1,2,3}}]+Sum[v[j]((e[j])/(1-((e[j])^2))),{j,{1,2,3}}].Matrix[C2,s].Sum[vb[j](2/(((1-((eb[j])^2)))^2)),{j,{1,2,3}}] -4Vector[Piota,s].Matrix[B,s].Matrix[C2,s].Vector[P,s]+Sum[v[j]((2 ((e[j])^2))/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].Matrix[C1,s].Sum[vb[j](1/(1-((eb[j])^2))),{j,{1,2,3}}]+Sum[v[j]((2 ((e[j])^2))/(1-((e[j])^2))),{j,{1,2,3}}].Matrix[C1,s].Sum[vb[j]((2/(((1-((eb[j])^2)))^2))+(2/(1-((eb[j])^2)))),{j,{1,2,3}}];
Bound[Xi,iota,OddTail,Delta,ei,s]=Vector[hII,s].Sum[vb[j]*((eb[j])^4)((2/(((1-((eb[j])^2)))^2))+(5/((1-((eb[j])^2))))),{j,{1,2,3}}]+Sum[((e[j])^4)((2/(((1-((e[j])^2)))^2))+(5/((1-((e[j])^2)))))*v[j],{j,{1,2,3}}].(Matrix[H3,s].Vector[P,s]+Matrix[Ai,s].Vector[h,s])+Sum[v[j]((2 e[j])/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].Matrix[C2,s].Sum[vb[j](1/(1-((eb[j])^2))),{j,{1,2,3}}]+Sum[v[j]((e[j])/(1-((e[j])^2))),{j,{1,2,3}}].Matrix[C2,s].Sum[vb[j](2/(((1-((eb[j])^2)))^2)),{j,{1,2,3}}] +Sum[v[j]((2 ((e[j])^2))/(((1-((e[j])^2)))^2)),{j,{1,2,3}}].Matrix[C1,s].Sum[vb[j](1/(1-((eb[j])^2))),{j,{1,2,3}}]+Sum[v[j]((2 ((e[j])^2))/(1-((e[j])^2))),{j,{1,2,3}}].Matrix[C1,s].Sum[vb[j]((2/(((1-((eb[j])^2)))^2))+(3/(1-((eb[j])^2)))),{j,{1,2,3}}]+Sum[((e[j])^3)((2/(((1-((e[j])^2)))^2))+(5/((1-((e[j])^2)))))*v[j],{j,{1,2,3}}].Matrix[Ai,s].Vector[P,s];
,{s,{i,o}}]

For Sum over all N

Do[
Do[
Bound[Xi,a, Even,Delta,0, s]=Sum[Bound[Xi,a,t,Delta,0, s],{t,{0,2,EvenTail}}];
Bound[Xi,a, Odd,Delta,0, s]=Sum[Bound[Xi,a,t,Delta,0, s],{t,{1,3,OddTail}}];
Bound[Xi,a, Absolut,Delta,0, s]=Sum[Bound[Xi,a, t,Delta,0, s],{t,{Odd,Even}}];
,{a,{normal,iota}}];

Bound[Xi,iota, Even,Delta,ei, s]=Sum[Bound[Xi,iota,t,Delta,ei, s],{t,{0,2,EvenTail}}];
Bound[Xi,iota, Odd,Delta,ei, s]=Sum[Bound[Xi,iota,t,Delta,ei, s],{t,{1,3,OddTail}}];
Bound[Xi,iota, Absolut,Delta,ei, s]=Sum[Bound[Xi,iota,t,Delta,ei, s],{t,{Odd,Even}}];
,{s,{i,o}}]

Computation of constants of Proposition 3.3.1

In this section we perform the computations of Section 3.4. to obtain the constances stated in Proposition 3.3.1:

(∑_x)|F(x)|≤  (Subscript[K, F])=Bound[KF]
Bound[KPhi,1]=(Subscript[(UnderBar[K]), Φ])≤(OverHat[Φ])(0)≤  (Subscript[(OverBar[K]), Φ])=Bound[KPhi,2]
Bound[KPhiabs,1]=(Subscript[(UnderBar[K]), |Φ|])≤(∑_x)|Φ(x)|≤  (Subscript[(OverBar[K]), |Φ|])=Bound[KPhiabs,2]
(∑_(x≠0))|(Subscript[Φ, z])(x)|≤  (Subscript[K, |Φ|])=Bound[KPhiWithoutZero]

(∑_x)F(x)[1-cos(k x)]≥ (Subscript[K, Lower])[1-(OverHat[D])(k)]
(∑_x)|F(x)|[1-cos(k x)]≤  (Subscript[K, ΔF])[1-(OverHat[D])(k)]
(∑_x)|(Subscript[Φ, z])(x)|[1-cos(k x)]≤  (Subscript[K, ΔΦ])[1-(OverHat[D])(k)]

Bound on absolute value (Subscript[K, F])and (Subscript[K, Φ])

Do[
alpha[s]=z[s] rho[s];
baralpha[s]=z[s];
Bound[KPsi,s]=rho[s]+((2d-2)/(2d))Bound[Xi,normal,Absolut,s];
Bound[KF,s]=(((2d baralpha[s]))/((1-alpha[s]-(2d-2)alpha[s]Bound[Xi,iota,Absolut,s])))Bound[KPsi,s];
Bound[KPhiup,s]=1+Bound[Xi,normal,Even,s]+Bound[Xi,iota,Absolut,s]Bound[KF,s];
Bound[KPhidown,s]=1-Bound[Xi,normal,Odd,s]-Bound[Xi,iota,Absolut,s]Bound[KF,s];
Bound[KPhiabsup,s]=1+Bound[Xi,normal,Absolut,s]+Bound[Xi,iota,Absolut,s]Bound[KF,s];
Bound[KPhiabsdown,s]=1-Bound[Xi,normal,Absolut,s]-Bound[Xi,iota,Absolut,s]Bound[KF,s];
Bound[KPhiWithoutZero,s]=Bound[Xi,normal,Absolut,s]+Bound[Xi,iota,Absolut,s]Bound[KF,s];
,{s,{i,o}}]

Bounds on differences

As next we implement the computation of Section 3.4.3. First the differences of (Subscript[F, 1])and (Subscript[Φ, 1]), lines (3.4.26),  (3.4.27),  (3.4.29)

Do[
Bound[DifferencefF,Part1,Lower,s]= Min[((2d baralpha[i])/(1-((alpha[i])^2))),((2d baralpha[s])/(1-((alpha[s])^2)))](rho[Lower,s]-Bound[Xi,normal,Odd,Delta,0,s]-Bound[Xi,normal,Odd,s]-alpha[o]Bound[Xi,normal,Even,Delta,0,s]);
Bound[DifferencefF,Part1,Absolut,s]=Max[((2d baralpha[i])/(1-((alpha[i])^2))),((2d baralpha[s])/(1-((alpha[s])^2)))](rho[s]+(1+alpha[s])Bound[Xi,normal,Absolut,Delta,0,s]+Bound[Xi,normal,Absolut,s]);
Bound[KDeltaPhi,Part1,s]=Bound[Xi,normal,Absolut,Delta,0,s]+((baralpha[s])/(1-((alpha[s])^2)))(2d Bound[Xi,normal,Absolut,Delta,0,s]Bound[Xi,iota,Absolut,s]+(1+Bound[Xi,normal,Absolut,s])Bound[Xi,iota,Absolut,Delta,ei,s]+2d alpha[s]Bound[Xi,normal,Absolut,Delta,0,s]Bound[Xi,iota,Absolut,s]+alpha[s](1+Bound[Xi,normal,Absolut,s])Bound[Xi,iota,Absolut,Delta,0,s]);
,{s,{i,o}}]

Then the differences of (Subscript[F, 2])and (Subscript[Φ, 2]): For the bound in (3.4.30), (3.4.32) and  (3.4.35)

Do[
Bound[DifferencefF,Part2,Lower,s]=
-((((2d baralpha[s]))^2)/(1-((alpha[s])^2)))(Bound[Xi,normal,Odd,Delta,0,s]Bound[Xi,iota,Odd,s]+Bound[Xi,normal,Even,Delta,0,s]Bound[Xi,iota,Even,s])
-((2d ((baralpha[s])^2))/(1-((alpha[s])^2)))(rho[s]+((2d-2)/(2d))Bound[Xi,normal,Even,s])( Bound[Xi,iota,Even,Delta,ei,s]+2d Bound[Xi,iota,Even,s]+ ((alpha[s])^2)Bound[Xi,iota,Even,Delta,0,s]+alpha[s](Bound[Xi,iota,Odd,Delta,ei,s]+Bound[Xi,iota,Odd,Delta,0,s]+2d Bound[Xi,iota,Odd,s]))-((2d ((baralpha[s])^2))/(1-((alpha[s])^2)))(((2d-2)/(2d))Bound[Xi,normal,Odd,s])( Bound[Xi,iota,Odd,Delta,ei,s]+2d Bound[Xi,iota,Odd,s]+ ((alpha[s])^2)Bound[Xi,iota,Odd,Delta,0,s]+alpha[s](Bound[Xi,iota,Even,Delta,ei,s]+Bound[Xi,iota,Even,Delta,0,s]+2d Bound[Xi,iota,Even,s]));
Bound[DifferencefF,Part2,Absolut,s]=((((2d baralpha[s]))^2)/(1-((alpha[s])^2)))Bound[Xi,normal,Absolut,Delta,0,s]Bound[Xi,iota,Absolut,s]
+((2d ((baralpha[s])^2))/((1-((alpha[s])^2))(1+alpha[s])))(rho[s]+((2d-2)/(2d))Bound[Xi,normal,Absolut,s])( Bound[Xi,iota,Absolut,Delta,ei,s]+2d Bound[Xi,iota,Absolut,s]+ alpha[s]Bound[Xi,iota,Absolut,Delta,0,s]);
Bound[KDeltaPhi,Part2,s]=((2d ((baralpha[s])^2))/((1-((alpha[s])^2))))(2d Bound[Xi,normal,Absolut,Delta,0,s] ((Bound[Xi,iota,Absolut,s])^2)+2(1+Bound[Xi,normal,Absolut,s])((Bound[Xi,iota,Absolut,s])/(1+alpha[s]))(Bound[Xi,iota,Absolut,Delta,ei,s]+alpha[s]Bound[Xi,iota,Absolut,Delta,0,s]));
,{s,{i,o}}]

Finally, we compute the differences of (Subscript[F, 3])and (Subscript[Φ, 3]), lines  (4.4.37) and  (4.4.38)

Do[
tmp=(1/(1-((2d alpha[s] Bound[Xi,iota,Absolut,s])/(1-alpha[s]))));
Bound[DifferencefF,Part3,Absolut,s]=Bound[Xi,normal,Absolut,Delta,0,s]((2d baralpha[s])/(((1-alpha[s]))^3))(((2d alpha[s]Bound[Xi,iota,Absolut,s]))^2)tmp+(1+Bound[Xi,normal,Absolut,s])((baralpha[s](((2d alpha[s]))^2))/((((1-alpha[s]))^2)(1-((alpha[s])^2))))Bound[Xi,iota,Absolut,s](tmp^2)(Bound[Xi,iota,Absolut,Delta,ei,s]+alpha[s] Bound[Xi,iota,Absolut,Delta,0,s])+(1+Bound[Xi,normal,Absolut,s])((baralpha[s](((2d alpha[s]))^2))/((((1-alpha[s]))^2)(1-((alpha[s])^2))))Bound[Xi,iota,Absolut,s]tmp(Bound[Xi,iota,Absolut,Delta,ei,s]+alpha[s] Bound[Xi,iota,Absolut,Delta,0,s]+2d Bound[Xi,iota,Absolut,s]);

Bound[DifferencefF,Part3,Lower,s]=-Bound[DifferencefF,Part3,Absolut,s];
Bound[KDeltaPhi,Part3,s]=Bound[Xi,normal,Absolut,Delta,0,s]((2d baralpha[s]Bound[Xi,iota,Absolut,s])/(((1-alpha[s]))^3))(((2d alpha[s]Bound[Xi,iota,Absolut,s]))^2)tmp+(1+Bound[Xi,normal,Absolut,s])(baralpha[s](((2d alpha[s]Bound[Xi,iota,Absolut,s]))^2))/((((1-alpha[s]))^2)(1-((alpha[s])^2)))((tmp^2)+tmp)(1/(1+alpha[s]))(Bound[Xi,iota,Absolut,Delta,ei,s]+alpha[s] Bound[Xi,iota,Absolut,Delta,0,s]);

Bound[KDeltaFLower,s]=1/(Bound[DifferencefF,Part1,Lower,s]+Bound[DifferencefF,Part2,Lower,s]+Bound[DifferencefF,Part3,Lower,s]);
Bound[KDeltaF,s]=Bound[DifferencefF,Part1,Absolut,s]+Bound[DifferencefF,Part2,Absolut,s]+Bound[DifferencefF,Part3,Absolut,s];
Bound[KDeltaPhi,s]=Bound[KDeltaPhi,Part1,s]+Bound[KDeltaPhi,Part2,s]+Bound[KDeltaPhi,Part3,s];
Clear[tmp];
,{s,{i,o}}]

Check of the sufficient condition

Now we can compute whether P(γ,Γ,z) is satisfied, see Definition 3.3.2.

Do[
NoBLEBoundF1[s]=((1+((2d-2)/(2d-1)) Gamma1 Bound[Xi,iota,Even,s])/(rho[s]-((2d-2)/(2d)) Bound[Xi,normal,Odd,s]));
NoBLEBoundF2[s]=((2d-2)/(2d-1))Bound[KPhiabsup,s]Bound[KDeltaFLower,s];
,{s,{i,o}}]

and we compute for (Subscript[f, 3])

Do[
NoBLEBoundF3[Part1,s] =(1/(2c1))Bound[KDeltaFLower,s]Bound[KDeltaPhi,s];
NoBLEBoundF3[Part2,s] =(1/(2c2))Bound[KPhiabsup,s]Bound[KDeltaF,s]((Bound[KDeltaFLower,s])^2);
NoBLEBoundF3[Part3,s]=2(((Bound[KDeltaFLower,s])^2)/c3)√(Bound[KDeltaF,s]Bound[KDeltaPhi,s]Bound[KPhiWithoutZero,s]Bound[KF,s]);
NoBLEBoundF3[Part4,s]=2(((Bound[KDeltaFLower,s])^2)/c4)(2Bound[KPhiabsup,s]((Bound[KDeltaF,s])^2)Bound[KDeltaFLower,s]+√(Bound[KDeltaF,s]Bound[KDeltaPhi,s]Bound[KPhiWithoutZero,s]Bound[KF,s]));
NoBLEBoundF3[s]=Max[NoBLEBoundF3[Part1,s],NoBLEBoundF3[Part2,s],NoBLEBoundF3[Part3,s],NoBLEBoundF3[Part4,s]];
,{s,{i,o}}]

We finally check

Do[
Succes[f1,s]=NoBLEBoundF1[s]<Gamma1;
Succes[f2,s]=NoBLEBoundF2[s]<Gamma2;
Succes[f3,s]=NoBLEBoundF3[s]<Gamma3;
Succes[s]=Succes[f1,s]&&Succes[f2,s]&& Succes[f3,s];
,{s,{i,o}}]
Succes[overall]=Succes[i]&&Succes[o];

Result

The overall result

The statement that the bootstrap was succesful is

Succes[overall]

If this succedes than the analysis of Section 3.3 can be used to proved mean-field behavoior for Percolation.

Assuming the bootstrap was succesful we prove the following bounds: The infrared bound in (3.3.12) and (3.3.13) hold with

((2d-2)/(2d-1))Gamma2(* ≥ (Subscript[G, z])(k)[1-(OverHat[D])(k)]*)
Max[Bound[KDeltaFLower,o],1](* Nominator in (4.3.13) *)

Further, we have proven that (Subscript[g, (Subscript[z, c])])(Subscript[z, c]) is smaller than

(1/(2d-1))Gamma1

The improvement of bounds

bubbles={Graphics[{Green,Disk[{0,0},0.8]},ImageSize-> 30],Graphics[{Red,Disk[{0,0},0.8]},ImageSize-> 40]};
tableClassicCheck={{Bounds,Init-(Subscript[f, 1]),Init-(Subscript[f, 2]),Init1-(Subscript[f, 3]),Init2-(Subscript[f, 3]),Init3-(Subscript[f, 3]),Init4-(Subscript[f, 3]),(Subscript[f, 1]),(Subscript[f, 2]),(Subscript[f, 31]),(Subscript[f, 32]),(Subscript[f, 33]),(Subscript[f, 34])},{Gamma,Gamma1,Gamma2,Gamma3 ,Gamma3 ,Gamma3 ,Gamma3 ,Gamma1,Gamma2,Gamma3,Gamma3 ,Gamma3 ,Gamma3  },{Bounds,NoBLEBoundF1[i],NoBLEBoundF2[i],NoBLEBoundF3[Part1,i] ,NoBLEBoundF3[Part2,i] ,NoBLEBoundF3[Part3,i],NoBLEBoundF3[Part4,i] ,NoBLEBoundF1[o],NoBLEBoundF2[o],NoBLEBoundF3[Part1,o] ,NoBLEBoundF3[Part2,o] ,NoBLEBoundF3[Part3,o] ,NoBLEBoundF3[Part4,o] },{ check,
If[NoBLEBoundF1[i]<Gamma1,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF2[i]<Gamma2,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF3[Part1,i]<Gamma3,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF3[Part2,i]<Gamma3,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF3[Part3,i]<Gamma3,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF3[Part4,i]<Gamma3,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF1[o]<Gamma1,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF2[o]<Gamma2,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF3[Part1,o]<Gamma3,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF3[Part2,o]<Gamma3,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF3[Part3,o]<Gamma3,bubbles[[1]],bubbles[[2]]],If[NoBLEBoundF3[Part4,o]<Gamma3,bubbles[[1]],bubbles[[2]]],},{Required,NoBLEBoundF1[i],NoBLEBoundF2[i],NoBLEBoundF3[Part1,i]/Gamma3 ,NoBLEBoundF3[Part2,i] /Gamma3,NoBLEBoundF3[Part3,i]*c3/Gamma3 ,NoBLEBoundF3[Part4,i]*c4 /Gamma3,NoBLEBoundF1[o],NoBLEBoundF2[o],NoBLEBoundF3[Part1,o]*c1/Gamma3 ,NoBLEBoundF3[Part2,o] *c2/Gamma3,NoBLEBoundF3[Part3,o]*c3/Gamma3,NoBLEBoundF3[Part4,o]*c4/Gamma3}};
Labeled[Grid[tableClassicCheck,Alignment->{Center},Frame->True,Dividers->{{2->True,-1->True},{2->True}},ItemStyle->{1->Bold,1->Bold},Background->{Automatic,Automatic,{{2,2},{2,15}}->GrayLevel[0.7]}],Style["Result for Dimension "Text[d],Bold],Top]//Text

Semi-automate proceedure to find appropiate value for the constants (Subscript[Γ, i])and (Subscript[c, i]).

Initially we guess a good value for the constant and make a first computation. Then we deactivate there initial definition in the top of the document and use the code below. We recompile the document a number of times and hope that the algorithm below converges to a fixed point.

(*d=38;
{d,Gamma1,Gamma2,Gamma3,c1,c2,c3,c4}
Gamma1=NoBLGamma1=NoBLEBoundF1[o]+0.000001;
Gamma2=NoBLEBoundF2[o]+0.000001;
c1=NoBLEBoundF3[Part1,o]*c1/Gamma3+0.00001;
c2=NoBLEBoundF3[Part2,o]*c2/Gamma3+0.00001;
c3=NoBLEBoundF3[Part3,o]*c3/Gamma3+0.00001;
c4=NoBLEBoundF3[Part4,o]*c4/Gamma3+0.0001;
{d,Gamma1,Gamma2,Gamma3,c1,c2,c3,c4}*)

Print out of the computed bounds in the coefficients

Do[MethodeFourTable[s]={{Quantity,(Ξ^Zero),(Ξ^One),(Ξ^Two),(Ξ^Three),(Ξ^EvenTail),(Ξ^OddTail)},{Text[Bound for (OverHat[Ξ])],Bound[Xi,normal,0,s],Bound[Xi,normal,1,s] ,Bound[Xi,normal,2,s],Bound[Xi,normal,3,s],Bound[Xi,normal,EvenTail,s],Bound[Xi,normal,OddTail,s]},{Text[Bound for ((OverHat[Ξ])^ι)],Bound[Xi,iota,0,s],Bound[Xi,iota,1,s] ,Bound[Xi,iota,2,s],Bound[Xi,iota,3,s],Bound[Xi,iota,EvenTail,s],Bound[Xi,iota,OddTail,s]},{Text[(OverHat[Ξ])(1-cos(kx))],Bound[Xi,normal,0,Delta,0,s],Bound[Xi,normal,1,Delta,0,s],Bound[Xi,normal,2,Delta,0,s],Bound[Xi,normal,3,Delta,0,s],Bound[Xi,normal,EvenTail,Delta,0,s],Bound[Xi,normal,OddTail,Delta,0,s]},{Text[ (Ξ^ι)(1-cos(kx))],Bound[Xi,iota,0,Delta,0,s],Bound[Xi,iota,1,Delta,0,s],Bound[Xi,iota,2,Delta,0,s],Bound[Xi,iota,3,Delta,0,s],Bound[Xi,iota,EvenTail,Delta,0,s],Bound[Xi,iota,OddTail,Delta,0,s]},{Text[(Ξ^ι)(1-cos(k(x-(Subscript[e, ι]))))] ,Bound[Xi,iota,0,Delta,ei,s],Bound[Xi,iota,1,Delta,ei,s] ,Bound[Xi,iota,2,Delta,ei,s],Bound[Xi,iota,3,Delta,ei,s],Bound[Xi,iota,EvenTail,Delta,ei,s],Bound[Xi,iota,OddTail,Delta,ei,s]}};
,{s,{i,o}}]
MethodeFourTablePart1={{Quantity,KF,KPhi,DELTAFLower,DELTAFAbsolut,DELTAPhi},{Bound for ,Bound[KF,o],Bound[KPhiup,o],Bound[KDeltaFLower,o],Bound[KDeltaF,o],Bound[KDeltaPhi,o]}};
MethodeFourTablePart2={{Quantity,DELTAFLower,2,3,DELTAFAbsolut,2,3,DELTAPhi,2,3},{Bound for ,Bound[DifferencefF,Part1,Lower,o],Bound[DifferencefF,Part2,Lower,o],Bound[DifferencefF,Part3,Lower,o],Bound[DifferencefF,Part1,Absolut,o],Bound[DifferencefF,Part2,Absolut,o],Bound[DifferencefF,Part3,Absolut,o],Bound[KDeltaPhi,Part1,o],Bound[KDeltaPhi,Part2,o],Bound[KDeltaPhi,Part3,o]}};

Labeled[Grid[MethodeFourTable[i],Alignment->{Center},Frame->True,Dividers->{{2->True,-1->True},{2->True}},ItemStyle->{1->Bold,1->Bold},Background->{{None},{GrayLevel[0.9]},{None}}],Style["Bound on coefficients at \!\(\*SubscriptBox[\(z\), \(\(I\)\(\\ \)\)]\)in Dimension "Text[d],Bold],Top]//Text
Labeled[Grid[MethodeFourTable[o],Alignment->{Center},Frame->True,Dividers->{{2->True,-1->True},{2->True}},ItemStyle->{1->Bold,1->Bold},Background->{{None},{GrayLevel[0.9]},{None}}],Style["Bound on coefficients in Dimension "Text[d],Bold],Top]//Text

Labeled[Grid[MethodeFourTablePart1,Alignment->{Center},Frame->True,Dividers->{{2->True,-1->True},{2->True}},ItemStyle->{1->Bold,1->Bold},Background->{{None},{GrayLevel[0.9]},{None}}],Style["Bound on the constants of Proposition 3.3.1 in Dimension "Text[d],Bold],Top]//Text
Labeled[Grid[MethodeFourTablePart2,Alignment->{Center},Frame->True,Dividers->{{2->True,-1->True},{2->True}},ItemStyle->{1->Bold,1->Bold},Background->{{None},{GrayLevel[0.9]},{None}}],Style["Bound on the constants of Proposition 3.3.1 in Dimension "Text[d],Bold],Top]//Text