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\def\titletesi{Universality and non--universality 
in the Ashkin--Teller model}

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%\centerline{\titolo{
%Universit\'a degli Studi di Roma \\ \vspace{-0.3cm}La Sapienza\\
%Facolt\'a di Scienze Matematiche Fisiche  \\
%\vspace{-0.3cm}Naturali }}
\centerline{\csss UNIVERSIT\`A DEGLI STUDI DI ROMA}
\vskip.05truecm
\centerline{\csss LA SAPIENZA}
\vskip1.truecm
\centerline{\cs FACOLT\`A DI SCIENZE MATEMATICHE, FISICHE E NATURALI}
\vskip1.5truecm
\centerline{\cs PhD thesis in Physics}
\centerline{ A.Y. 2003/2004}
\vskip1.5truecm
\centerline{\titolone Universality and non--universality}
\vskip.05truecm
\centerline{\titolone in the Ashkin--Teller model}
\prtindex{Universality and non--universality
in the Ashkin--Teller model}
\vskip2.truecm

\leftline{\css Advisors:}
\leftline{{\css Prof.}{\it Giovanni Gallavotti}}
\leftline{{\css Prof.}{\it Vieri Mastropietro}}
\vskip1.truecm
\leftline{\css Author:}
\leftline{\it Alessandro Giuliani}
\ppagina
\null
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\centerline{\titolo Index:}
\acapo\acapo
\hbox to 6.3 in{\bf 1. Introduction.\dotfill 7}\\
\hbox to 6.3 in{\qquad 1.1. The Ashkin--Teller model.\dotfill 9}\\
\hbox to 6.3 in{\qquad 1.2. Results.\dotfill 11}\\
\hbox to 6.3 in{\qquad 1.3. Outline of the proof.\dotfill 14}\\
\hbox to 6.3 in{\qquad 1.4. Summary.\dotfill 16}\\
\hbox to 6.3 in{\bf 2. The Ising model exact solution.\dotfill 18}\\
\hbox to 6.3 in{\qquad 2.1. The multipolygon representation.\dotfill 18}\\
\hbox to 6.3 in{\qquad 2.2. The Grassmann integration rules.\dotfill 18}\\
\hbox to 6.3 in{\qquad 2.3. The Grassmann representation of the 2D Ising model 
with open boundary conditions.\dotfill 19}\\
\hbox to 6.3 in{\qquad 2.4. The Grassmann representation of the 2D Ising model 
with periodic boundary conditions.\dotfill 27}\\
\hbox to 6.3 in{\qquad 2.5. The Ising model's free energy.\dotfill 28}\\
\hbox to 6.3 in{\bf 3. The Grassmann formulation of Ashkin--Teller.\dotfill 30}\\
\hbox to 6.3 in{\qquad 3.1. The Grassmann representation for a pair of 
Ising models with multi spin interactions.\dotfill 30}\\
\hbox to 6.3 in{\qquad 3.2. The Grassmann representation for the 
Ashkin--Teller model.\dotfill 33}\\
\hbox to 6.3 in{\bf 4. The ultraviolet integration.\dotfill 37}\\
\hbox to 6.3 in{\qquad 4.1. The effective interaction on scale 1.\dotfill 37}\\
\hbox to 6.3 in{\qquad 4.2. The integration of the $\c$ fields.\dotfill 38}\\
\hbox to 6.3 in{\qquad 4.3. Symmetry properties.\dotfill 43}\\
\hbox to 6.3 in{\bf 5. Renormalization Group for light fermions. 
The anomalous regime.\dotfill 48}\\
\hbox to 6.3 in{\qquad 5.1. Multiscale analysis.\dotfill 48}\\
\hbox to 6.3 in{\qquad 5.2. The localization operator.\dotfill 49}\\
\hbox to 6.3 in{\qquad 5.3. Renormalization.\dotfill 51}\\
\hbox to 6.3 in{\qquad 5.4. Analyticity of the effective potential.\dotfill 53}\\
\hbox to 6.3 in{\qquad 5.4. Proof of Theorem 5.1.\dotfill 54}\\
\hbox to 6.3 in{\bf 6. The flow of the running coupling constants.\dotfill 61}\\
\hbox to 6.3 in{\qquad 6.1. The flow equations.\dotfill 61}\\
\hbox to 6.3 in{\qquad 6.2. Proof of Theorem 6.1.\dotfill 61}\\
\hbox to 6.3 in{\qquad 6.2. The flow of the renormalization constants.\dotfill 64}\\
\hbox to 6.3 in{\qquad 6.4. The scale $h^*_1$.\dotfill 65}\\
\hbox to 6.3 in{\bf 7. Renormalization Group for light fermions. 
The non anomalous regime.\dotfill 67}\\
\hbox to 6.3 in{\qquad 7.1. Integration of the $\psi^{(1)}$ field.\dotfill 67}\\
\hbox to 6.3 in{\qquad 7.2. The localization operator.\dotfill 68}\\
\hbox to 6.3 in{\qquad 7.3. Renormalization for $h\le h^*_1$.\dotfill 69}\\
\hbox to 6.3 in{\qquad 7.4. The integration of the scales $\le h^*_2$.\dotfill 70}\\
\hbox to 6.3 in{\qquad 7.5. Keeping $h^*_2$ finite.\dotfill 71}\\
\hbox to 6.3 in{\qquad 7.6. The critical points.\dotfill 71}\\
\hbox to 6.3 in{\qquad 7.7. Computation of $h^*_2$.\dotfill 72}\\
\hbox to 6.3 in{\bf 8. The specific heat.\dotfill 74}\\
\hbox to 6.3 in{\bf 9. Conclusions and open problems.\dotfill 78}\\
\hbox to 6.3 in{\bf Appendix A1. Grassmann integration. Truncated expectations.
\dotfill 79}\\
\hbox to 6.3 in{\qquad A1.1 Truncated expectations and some more rules.\dotfill 79}\\
\hbox to 6.3 in{\qquad A1.2 Graphical representation for truncated expectations.\dotfill
81}\\ 
\hbox to 6.3 in{\bf Appendix A2. The Pfaffian expansion.\dotfill 83}\\
\hbox to 6.3 in{\bf Appendix A3. Gram--Hadamard inequality.\dotfill 90}\\
\hbox to 6.3 in{\bf Appendix A4. Proof of Lemma 5.3.\dotfill 94}\\
\hbox to 6.3 in{\bf Appendix A5. Proof of {(5.42)}.\dotfill 95}\\
\hbox to 6.3 in{\bf Appendix A6. Vanishing of the Beta function.\dotfill 96}\\
\hbox to 6.3 in{\qquad A6.1. The reference model.\dotfill 96}\\
\hbox to 6.3 in{\qquad A6.2. The Dyson equation.\dotfill 97}\\
\hbox to 6.3 in{\qquad A6.3. Ward identities and the first addend of 
\equ(A6.2.11a).\dotfill 99}\\
\hbox to 6.3 in{\qquad A6.4. The first correction identity.\dotfill 102}\\
\hbox to 6.3 in{\qquad A6.5. Ward identities and the second addend in 
\equ(A6.2.11a).\dotfill 108}\\
\hbox to 6.3 in{\qquad A6.6. Proof of \equ(A6.2.26).\dotfill 112}\\
\hbox to 6.3 in{\bf Appendix A7. The properties of 
{$D_\o ({\fam \bffam \tenbf p})^{-1} C_\o ({\fam \bffam \tenbf k},
{\fam \bffam \tenbf k}-{\fam \bffam \tenbf p}) $}.\dotfill 127}\\
\hbox to 6.3 in{\bf Appendix A8. Proof of Lemma 7.3.\dotfill 129}\\
%\hbox to 6.3 in{\qquad A8.1 Tree expansion for $\b^h_\p$.\dotfill 120}\\
%\hbox to 6.3 in{\qquad A8.2 Bounds on $\b^h_\p$.\dotfill 120}\\
%\hbox to 6.3 in{\qquad A8.3 Fixing the counterterm.\dotfill 120}\\
%\hbox to 6.3 in{\qquad A8.4 The flows of $Z_h$ and $\widehat m^{(2)}_h$.
%\dotfill 120}\\
%\hbox to 6.3 in{\qquad A8.5 The Lipshitz property \equ(7.ph).\dotfill 120}\\
\hbox to 6.3 in{\bf Appendix A9. Independence from boundary conditions.\dotfill
131}\\
\hbox to 6.3 in{\bf Acknowledgments.\dotfill 132}\\
\hbox to 6.3 in{\bf References.\dotfill 133}\\ 

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 01 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{1. Introduction.}
\capindex{1}{Introduction.}
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\input cap1.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 02 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{2. The Ising model exact solution.}
\capindex{2}{The Ising model exact solution.}
\vskip1.truecm
\input cap2.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 03 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{3. The Grassmann formulation of Ashkin--Teller.}
\capindex{3}{The Grassmann formulation of Ashkin--Teller.}
\vskip1.truecm
\input cap3.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 04 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{4. The ultraviolet integration.}
\capindex{4}{The ultraviolet integration.}
\vskip1.truecm
\input cap4.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 05 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{5. Renormalization Group for light fermions. The anomalous regime.}
\capindex{5}{Renormalization Group for light fermions. The anomalous regime.}
\vskip1.truecm
\input cap5.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 06 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{6. The flow of the running coupling constants.}
\capindex{6}{The flow of the running coupling constants.}
\vskip1.truecm
\input cap6.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 07 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{7. Renormalization Group for light fermions. The non anomalous regime.}
\capindex{7}{Renormalization Group for light fermions. 
The non anomalous regime.}
\vskip1.truecm
\input cap7.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 08 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{8. The specific heat.}
\capindex{8}{The specific heat.}
\vskip1.truecm
\input cap8.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Capitolo 09 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{9. Conclusions and open problems.}
\capindex{9}{Conclusions and open problems.}
\vskip1.truecm
\input cap9.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a1: Grassmann integration. Truncated expectations.}
\appindex{A1}{Grassmann integration. Truncated expectations.}
\vskip1.truecm
\input app1.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a2: The Pfaffian expansion.}
\appindex{A2}{The Pfaffian expansion.}
\vskip1.truecm
\input app2.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a3: Gram--Hadamard inequality.}
\appindex{A3}{Gram--Hadamard inequality.}
\vskip1.truecm
\input app3.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a4: Proof of Lemma 5.3.}
\appindex{A4}{Proof of Lemma 5.3.}
\vskip1.truecm
\input app4.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a5: Proof of {\equ(5.53)}.}
\appindex{A5}{Proof of {\equ(5.53)}.}
\vskip1.truecm
\input app5.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A6 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a6: Vanishing of the Beta function.}
\appindex{A6}{Vanishing of the Beta function.}
\vskip1.truecm
\input app6.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A7 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a7: The properties of {$D_\o(\pp)^{-1}
C_\o(\kk,\kk-\pp) $}.}
\appindex{A7}{The properties of {$D_\o(\pp)^{-1}
C_\o(\kk,\kk-\pp) $}.}
\vskip1.truecm
\input app7.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A8 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a8: Proof of Lemma 7.3.}
\appindex{A8}{Proof of Lemma 7.3.}
\vskip1.truecm
\input app8.txt
\pagina
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Appendice A9 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{appendix a9: Independence from boundary conditions.}
\appindex{A9}{Independence from boundary conditions.}
\vskip1.truecm
\input app9.txt
\pagina
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%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ringraziamenti %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcap{Acknowledgments}
\bibindex{Acknowledgments}
\vskip1.truecm
\centerline{\titolo Acknowledgments}
\acapo\acapo
I would like to thank
Prof. J. L. Lebowitz for his nice invitations to visit Rutgers University
in the last two years. Part of the work I presented here
was done in the stimulating atmosphere of Rutgers University.\\
\\ 
Vorrei ringraziare il Prof. G. Gallavotti e il Prof. G. Benfatto
per la fiducia, la disponibilita' e l'interesse 
con cui mi hanno seguito in questi anni.\\
Ringrazio  
il Prof. G. Gentile e il Prof. F. Bonetto, con i quali ho collaborato 
durante tutto il corso del Dottorato, e che hanno contribuito moltissimo
alla mia formazione scientifica.\\
Un particolare ringraziamento va infine al Prof. V. Mastropietro:
gli ultimi tre anni di stretta collaborazione con lui sono stati per me  
importantissimi. Senza le sue idee ed il suo fondamentale contributo 
non sarebbe stato possibile realizzare questo lavoro. 

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\setcap{References}
\bibindex{References}
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\centerline{\titolo References}
\input biblio.txt
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