12 0 obj Each node lof a ddimensional hypercubic lattice contains a spin ˙ l that can take sdi erent values, ˙ l 2f1;:::;sgwhere sis an integer greater or equal to 2. This is a tutorial review on the Potts model aimed at bringing out in an organized fashion the essential and important properties of the standard Potts model. << /Filter /FlateDecode /S 63 /Length 103 >> << /Pages 48 0 R /Type /Catalog >> A mathematically precise deﬁnition of the jump term ∥ u∥0 is rather technical in a spatially continuous setting. The Potts model may be used to “examine some of the individual incentives, and perceptions of difference, that can lead collectively to segregation …”. x��[Y��6�~ׯ�ۂ�"���Cc�,�. The relation between Potts model and percolation has been explained well in [3, 4]. �%WB��#|c}C��k.� ���9 �PP�I���%��ILH�� 82B44 1 Introduction One of the fundamental models in statistical physics is the nearest neighbor q-state Potts model. stream 0000007121 00000 n @Nr.��g���K>W @� �, � 82B44 1 Introduction One of the fundamental models in statistical physics is the nearest neighbor q-state Potts model. 0000002478 00000 n Potts model[1] and its connection to the percolation problem has been studied widely[2, 3]. The Potts model is de ned as follows. There have been wide studies of duality of Potts model and relation of Potts model to so many other models. The infinite-range Potts model is known as the Kac model. It has been a subject of intensive research since its introduction by Blatt et al in 1996. �ꇆ��n���Q�t�}MA�0�al������S�x ��k�&�^���>�0|>_�'��,�G! The variational formulation of the Potts model is given by arg min . Potts model, described in III and IV, can be used to derive percolation phase transitions. One way to tackle this NP-hard problem was proposed by Kov-tun [20, 21]. << /Filter /FlateDecode /Length 5095 >> <]>> 0 endobj %PDF-1.4 %���� Potts Model and Generalizations: Exact Results and Statistical Physics by Yan Xu Doctor of Philosophy in Physics Stony Brook University 2012 The q-state Potts model is a spin model that has been of longstand-ing interest as a many body system in statistical mechanics. 145 0 obj <> endobj 13 0 obj 0000001439 00000 n 2 Das Potts Modell An erster Stelle soll das Potts-Modell kurz in der Form angesprochen werden, in der es urspr¨unglich formuliert war. endstream endobj 154 0 obj<>stream We do … 0000001722 00000 n The Potts model is related to, and generalized by, several other models, including the XY model, the Heisenberg model and the N-vector model. the Potts model Marcelo Blatt, Shai Wiseman and Eytan Domany Department of Physics of Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel Abstract A new approach for clustering is proposed. 162 0 obj<>stream "F\$H:R��!z��F�Qd?r9�\A&�G���rQ��h������E��]�a�4z�Bg�����E#H �*B=��0H�I��p�p�0MxJ\$�D1��D, V���ĭ����KĻ�Y�dE�"E��I2���E�B�G��t�4MzN�����r!YK� ���?%_&�#���(��0J:EAi��Q�(�()ӔWT6U@���P+���!�~��m���D�e�Դ�!��h�Ӧh/��']B/����ҏӿ�?a0n�hF!��X���8����܌k�c&5S�����6�l��Ia�2c�K�M�A�!�E�#��ƒ�d�V��(�k��e���l ����}�}�C�q�9 �V��)g�B�0�i�W��8#�8wթ��8_�٥ʨQ����Q�j@�&�A)/��g�>'K�� �t�;\�� ӥ\$պF�ZUn����(4T�%)뫔�0C&�����Z��i���8��bx��E���B�;�����P���ӓ̹�A�om?�W= �i~�KJ���\$���3�C~lS�� ��Q��M��!�0]}�h��kd��s`�# 5zպ *5�jh�����΍Eys�J��A�^�A 0000002555 00000 n H�|R�n�0��+���|KdN} %���� trailer 0000002821 00000 n 0000001573 00000 n �x������- �����[��� 0����}��y)7ta�����>j���T�7���@���tܛ�`q�2��ʀ��&���6�Z�L�Ą?�_��yxg)˔z���çL�U���*�u�Sk�Se�O4?׸�c����.� � �� R� ߁��-��2�5������ ��S�>ӣV����d�`r��n~��Y�&�+`��;�A4�� ���A9� =�-�t��l�`;��~p���� �Gp| ��[`L��`� "A�YA�+��Cb(��R�,� *�T�2B-� 0000006810 00000 n The Hessian of Z M is the matrix H Z M pwq B2Z M Bw i j n i;j 0: When wPRn 1 ¡0, the largest eigenvalue of H Z M is simple and positive by the Perron-Frobenius theorem. endobj endstream 0000000016 00000 n 0P��*3�= Emphasis is placed on exact and rigorous results, but other aspects of the problem are also described to achieve a unified perspective. An e cient Cellular Potts Model algorithm that forbids cell fragmentation Marc Durand, Etienne Guesnet Matière et Systèmes Complexes (MSC), UMR 7057 CNRS & Université Paris Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France. endstream x�c```f``�d`a`0X� � `6+���a�b'� .�Su{kx��@E��30�1�30�s�1�+f�;!���u\$�P��n���T��� ��Y 0000003332 00000 n endstream endobj 155 0 obj<> endobj 156 0 obj<> endobj 157 0 obj<>stream 0000002220 00000 n It identiﬁes a part of an optimal solution by … endobj 0000003898 00000 n %PDF-1.5 x�b```f``z��\$�@�� Y83�800�I0p�Fv0D,��@J�qA�:&�V���N'f�}�ɤ}��p���#\$&00| M��,�h�\$粏�d�N��~�EgL�8(y�\$u\n1R�=�&��4u�p#�&�zCd�NI=��r� h�"�Qɔ���!�y-�pLbu���y}�,K>�`豛zŪf�0���җ�, � �N�D��9O\$S�~^��G8;���*WD��eÊ "�R�^�d��4'&rs�ç�s�nX Ի�d�Z%�5/����1��̋ ���aȤ�8k�Ŵʔ v��ӄΘ��0�#�S��IFsrg(���&�} N'��)�].�u�J�r� (1)u γ ∥∥ uAuf0 +∥ −∥ 2 2 Here, A is a linear operator (e.g., the Radon transform) and f is an element of the data space (e.g., a sinogram). M agrees with the partition function of the q-state Potts model, or the random cluster model [Pem00,Sok05,Gri06]. 0000006866 00000 n _�27*�T�I?�9Ni��O�:�����d"��JC�Pzo��SE��G ��3�*�h�;����qNi�������V����B�-S[��U��mn�����3�C�T�Ԟ/����~Mů�[�qD�lk�b5��p+z�}K�Q4J���Rl �)�q���������[�f��:{sH��P<=��˂��^�S�HG35���-��K���)�߀ut'�(w�pI���mמ��w�:�y�9[��UبWGڒ��� I �. << /Linearized 1 /L 214318 /H [ 1145 182 ] /O 14 /E 180326 /N 3 /T 213991 >> ��Vi&�ͼ �@�V #��DG�j �2-�����"@L�Q!�����`9Aa� . 0000001355 00000 n Potts model clustering is also known as the superparamagnetic clustering method. endstream endobj 146 0 obj<> endobj 147 0 obj<> endobj 148 0 obj<>/Font<>/ProcSet[/PDF/Text]/ExtGState<>/Pattern<>>> endobj 149 0 obj<> endobj 150 0 obj[/ICCBased 157 0 R] endobj 151 0 obj<> endobj 152 0 obj<> endobj 153 0 obj<>stream