3.6 Scaling limit for the sandpile height one eld
3.6.3 Proof of Theorem 5
Finally, we show Theorem 5.
Proof of Theorem 5. Let n ∈ N and for all 1 ≤ i ≤ n let fi ∈ Cc∞(U) and ti ∈ R. In
Proposition 6 (V is well dened) it is proven 0 < V = P
v∈Z2Cov(h0(0), h0(v)) < ∞. Therefore, the family of random variables fi hU, 1 ≤ i ≤ n, is well dened. We
write f := Pn i=1tifi, and note P 1≤i≤nti ·(fihU) = f hU and f ∈ C ∞ c (U). From Propositions 8 (The covariance matrix) and 7 (Higher cumulants vanish) as tends to
zero the cumulants of fhU converge to the cumulants of the normal distribution with
mean zero and variance R
Uf
2(z)dz. This is equivalent to convergence of the moments which in turn implies convergence in distribution. From Proposition 8 for all1≤i, j≤n
as→0 the covariance offihU and fjhU tends to R
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Acknowledgements
I thank Antal A. Járai and Franz Merkl; Antal A. Járai for supervision during my stay at Carleton University Ottawa, and Franz Merkl for supervising my Ph.D. thesis at Ludwig-Maximilians-Universität München.
Lebenslauf
Persönliche Daten:Name Florian Maximilian Dürre geboren am 27.04.1982 in München Familienstand ledig
Ausbildung:
1988 - 1992 Grundschule Riemerling / Feldbergschule München; 1992 - 2001 Gymnasium Ottobrunn
06/2001 Abitur
2001 - 2006 Diplomstudium Mathematik mit Nebenfach Informatik an der Ludwig-Maximilians-Universität München
05/2006 Mathematikdiplom Promotion:
seit 06/2006 Promotion an der Ludwig-Maximilians-Universität München bei Prof. Dr. F. Merkl
seit 06/2006 Wissenschaftlicher Mitarbeiter am Mathematischen Institut der Ludwig-Maximilians-Universität München
06/2007 - 03/2008 Forschungsaufenthalt an der Carleton University Ottawa bei Prof. Dr. A. A. Járai