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51. A Gibbsian model for message routeing in highly dense multihop networks

Author

König, Wolfgang and Tóbiás, András

Subjects
Mathematics - Probability, 60F10, 60G55, 60K30, 82B21, 90B15
Abstract

We investigate a probabilistic model for routeing of messages in relay-augmented multihop ad-hoc networks, where each transmitter sends one message to the origin. Given the (random) transmitter locations, we weight the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured in terms of the number of pairs of hops using the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of an optimization of the individual trajectories. In the limit of high spatial density of users, we describe the totality of all the message trajectories in terms of empirical measures. Employing large deviations arguments, we derive a characteristic variational formula for the limiting free energy and analyse the minimizer(s) of the formula, which describe the most likely shapes of the trajectory flow. The empirical measures of the message trajectories well describe the interference, but not the congestion; the latter requires introducing an additional empirical measure. Our results remain valid under replacing the two penalization terms by more general functionals of these two empirical measures., Comment: 40 pages

Published
2017

54. Global perspective of familial hypercholesterolaemia: a cross-sectional study from the EAS Familial Hypercholesterolaemia Studies Collaboration (FHSC)

Author

Vallejo-Vaz, Antonio J., Stevens, Christophe A.T., Lyons, Alexander R.M., Dharmayat, Kanika I., Freiberger, Tomas, Hovingh, G. Kees, Mata, Pedro, Raal, Frederick J., Santos, Raul D., Soran, Handrean, Watts, Gerald F., Abifadel, Marianne, Aguilar-Salinas, Carlos A., Alhabib, Khalid F., Alkhnifsawi, Mutaz, Almahmeed, Wael, Alnouri, Fahad, Alonso, Rodrigo, Al-Rasadi, Khalid, Al-Sarraf, Ahmad, Al-Sayed, Nasreen, Araujo, Francisco, Ashavaid, Tester F., Banach, Maciej, Béliard, Sophie, Benn, Marianne, Binder, Christoph J., Bogsrud, Martin P., Bourbon, Mafalda, Chlebus, Krzysztof, Corral, Pablo, Davletov, Kairat, Descamps, Olivier S., Durst, Ronen, Ezhov, Marat, Gaita, Dan, Genest, Jacques, Groselj, Urh, Harada-Shiba, Mariko, Holven, Kirsten B., Kayikcioglu, Meral, Khovidhunkit, Weerapan, Lalic, Katarina, Latkovskis, Gustavs, Laufs, Ulrich, Liberopoulos, Evangelos, Lima-Martinez, Marcos M., Lin, Jie, Maher, Vincent, Marais, A. David, März, Winfried, Mirrakhimov, Erkin, Miserez, André R., Mitchenko, Olena, Nawawi, Hapizah, Nordestgaard, Børge G., Panayiotou, Andrie G., Paragh, György, Petrulioniene, Zaneta, Pojskic, Belma, Postadzhiyan, Arman, Raslova, Katarina, Reda, Ashraf, Reiner, Željko, Sadiq, Fouzia, Sadoh, Wilson Ehidiamen, Schunkert, Heribert, Shek, Aleksandr B., Stoll, Mario, Stroes, Erik, Su, Ta-Chen, Subramaniam, Tavintharan, Susekov, Andrey V., Tilney, Myra, Tomlinson, Brian, Truong, Thanh Huong, Tselepis, Alexandros D., Tybjærg-Hansen, Anne, Vázquez Cárdenas, Alejandra, Viigimaa, Margus, Wang, Luya, Yamashita, Shizuya, Kastelein, John J.P., Bruckert, Eric, Vohnout, Branislav, Schreier, Laura, Pang, Jing, Ebenbichler, Christoph, Dieplinger, Hans, Innerhofer, Reinhold, Winhofer-Stöckl, Yvonne, Greber-Platzer, Susanne, Krychtiuk, Konstantin, Speidl, Walter, Toplak, Hermann, Widhalm, Kurt, Stulnig, Thomas, Huber, Kurt, Höllerl, Florian, Rega-Kaun, Gersina, Kleemann, Lucas, Mäser, Martin, Scholl-Bürgi, Sabine, Säly, Christoph, Mayer, Florian J., Sablon, Gaelle, Tarantino, Eric, Nzeyimana, Charlotte, Pojskic, Lamija, Sisic, Ibrahim, Nalbantic, Azra D., Jannes, Cinthia E., Pereira, Alexandre C., Krieger, Jose E., Petrov, Ivo, Goudev, Assen, Nikolov, Fedya, Tisheva, Snejana, Yotov, Yoto, Tzvetkov, Ivajlo, Baass, Alexis, Bergeron, Jean, Bernard, Sophie, Brisson, Diane, Brunham, Liam R., Cermakova, Lubomira, Couture, Patrick, Francis, Gordon A., Gaudet, Daniel, Hegele, Robert A., Khoury, Etienne, Mancini, G.B. John, McCrindle, Brian W., Paquette, Martine, Ruel, Isabelle, Cuevas, Ada, Asenjo, Sylvia, Wang, Xumin, Meng, Kang, Song, Xiantao, Yong, Qiang, Jiang, Tao, Liu, Ziyou, Duan, Yanyu, Hong, Jing, Ye, Pucong, Chen, Yan, Qi, Jianguang, Liu, Zesen, Li, Yuntao, Zhang, Chaoyi, Peng, Jie, Yang, Ya, Yu, Wei, Wang, Qian, Yuan, Hui, Cheng, Shitong, Jiang, Long, Chong, Mei, Jiao, Jian, Wu, Yue, Wen, Wenhui, Xu, Liyuan, Zhang, Ruiying, Qu, Yichen, He, Jianxun, Fan, Xuesong, Wang, Zhenjia, Chow, Elaine, Pećin, Ivan, Perica, Dražen, Symeonides, Phivos, Vrablik, Michal, Ceska, Richard, Soska, Vladimir, Tichy, Lukas, Adamkova, Vera, Franekova, Jana, Cifkova, Renata, Kraml, Pavel, Vonaskova, Katerina, Cepova, Jana, Dusejovska, Magdalena, Pavlickova, Lenka, Blaha, Vladimir, Rosolova, Hana, Nussbaumerova, Barbora, Cibulka, Roman, Vaverkova, Helena, Cibickova, Lubica, Krejsova, Zdenka, Rehouskova, Katerina, Malina, Pavel, Budikova, Milena, Palanova, Vaclava, Solcova, Lucie, Lubasova, Alena, Podzimkova, Helena, Bujdak, Juraj, Vesely, Jiri, Jordanova, Marta, Salek, Tomas, Urbanek, Robin, Zemek, Stanislav, Lacko, Jan, Halamkova, Hana, Machacova, Sona, Mala, Sarka, Cubova, Eva, Valoskova, Katerina, Burda, Lukas, Bendary, Ahmed, Daoud, Ihab, Emil, Sameh, Elbahry, Atef, Rafla, Samir, Sanad, Osama, Kazamel, Ghada, Ashraf, Mohamed, Sobhy, Mohamed, El-Hadidy, Amro, Shafy, Mohamed A., Kamal, Saif, Bendary, Mohamed, Talviste, Grete, Angoulvant, Denis, Boccara, Franck, Cariou, Bertrand, Carreau, Valérie, Carrie, Alain, Charrieres, Sybil, Cottin, Yves, Di-Fillipo, Mathilde, Ducluzeau, Pierre H., Dulong, Sonia, Durlach, Vincent, Farnier, Michel, Ferrari, Emile, Ferrieres, Dorota, Ferrieres, Jean, Gallo, Antonio, hankard, Regis, Inamo, Jocelyne, Lemale, Julie, Moulin, Philippe, Paillard, François, Peretti, Noel, Perrin, Agnès, Pradignac, Alain, Rabes, Jean P., Rigalleau, Vincent, Sultan, Ariane, Schiele, François, Tounian, Patrick, Valero, René, Verges, Bruno, Yelnik, Cécile, Ziegler, Olivier, Haack, Ira A., Schmidt, Nina, Dressel, Alexander, Klein, Isabel, Christmann, Jutta, Sonntag, Antonia, Stumpp, Christine, Boger, Diana, Biedermann, Dana, Usme, Monica M.N., Beil, F. Ulrich, Klose, Gerald, König, Christel, Gouni-Berthold, Ioanna, Otte, Britta, Böll, Gereon, Kirschbaum, Anja, Merke, Jürgen, Scholl, Johannes, Segiet, Thomas, Gebauer, Marco, Predica, Florentina, Mayer, Manfred, Leistikow, Frank, Füllgraf-Horst, Sabine, Müller, Cornelius, Schüler, Melanie, Wiener, Judith, Hein, Konrad, Baumgartner, Peter, Kopf, Stefan, Busch, Reinhold, Schömig, Michael, Matthias, Stephan, Allendorf-Ostwald, Nicole, Fink, Bruno, Böhm, Dieter, Jäkel, Alexander, Koschker, Ann-Cathrin, Schweizer, Rüdiger, Vogt, Anja, Parhofer, Klaus, König, Wolfgang, Reinhard, Wibke, Bäßler, Andrea, Stadelmann, Alexander, Schrader, Volker, Katzmann, Julius, Tarr, Adrienne, Steinhagen-Thiessen, Elisabeth, Kassner, Ursula, Paulsen, Gerret, Homberger, Jürgen, Zemmrich, Claudia, Seeger, Wolfgang, Biolik, Kathrin, Deiss, Dorothee, Richter, Corinna, Pantchechnikova, Elina, Dorn, Elena, Schatz, Ulrike, Julius, Ulrich, Spens, Antje, Wiesner, Tobias, Scholl, Michael, Rizos, Christos V., Sakkas, Nikolaos, Elisaf, Moses, Skoumas, Ioannis, Tziomalos, Konstantinos, Rallidis, Loukianos, Kotsis, Vasileios, Doumas, Michalis, Athyros, Vasileios, Skalidis, Emmanouil, Kolovou, Genovefa, Garoufi, Anastasia, Bilianou, Eleni, Koutagiar, Iosif, Agapakis, Dimitrios, Kiouri, Estela, Antza, Christina, Katsiki, Niki, Zacharis, Evangelos, Attilakos, Achilleas, Sfikas, George, Koumaras, Charalambos, Anagnostis, Panagiotis, Anastasiou, Georgia, Liamis, George, Koutsogianni, Amalia-Despoina, Karányi, Zsolt, Harangi, Mariann, Bajnok, László, Audikovszky, Mária, Márk, László, Benczúr, Béla, Reiber, István, Nagy, Gergely, Nagy, András, Reddy, Lakshmi L., Shah, Swarup A.V., Ponde, Chandrashekhar K., Dalal, Jamshed J., Sawhney, Jitendra P.S., Verma, Ishwar C., Altaey, Mays, Al-Jumaily, Khalid, Rasul, Dilshad, Abdalsahib, Ali F., Jabbar, Amer A., Al-ageedi, Mohanad, Agar, Ruth, Cohen, Hofit, Ellis, Avishay, Gavishv, Dov, Harats, Dror, Henkin, Yaacov, Knobler, Hila, Leavit, Leah, Leitersdorf, Eran, Rubinstein, Ardon, Schurr, Daniel, Shpitzen, Shoshi, Szalat, Auryan, Casula, Manuela, Zampoleri, Veronica, Gazzotti, Marta, Olmastroni, Elena, Sarzani, Riccardo, Ferri, Claudio, Repetti, Elena, Sabbà, Carlo, Bossi, Antonio Carlo, Borghi, Claudio, Muntoni, Sandro, Cipollone, Francesco, Purrello, Francesco, Pujia, Arturo, Passaro, Angelina, Marcucci, Rossella, Pecchioli, Valerio, Pisciotta, Livia, Mandraffino, Giuseppe, Pellegatta, Fabio, Mombelli, Giuliana, Branchi, Adriana, Fiorenza, Anna Maria, Pederiva, Cristina, Werba, Josè Pablo, Parati, Gianfranco, Carubbi, Francesca, Iughetti, Lorenzo, Iannuzzi, Arcangelo, Iannuzzo, Gabriella, Calabrò, Paolo, Averna, Maurizio, Biasucci, Giacomo, Zambon, Sabina, Roscini, Anna Rita, Trenti, Chiara, Arca, Marcello, Federici, Massimo, Del Ben, Maria, Bartuli, Andrea, Giaccari, Andrea, Pipolo, Antonio, Citroni, Nadia, Guardamagna, Ornella, Bonomo, Katia, Benso, Andrea, Biolo, Gianni, Maroni, Lorenzo, Lupi, Alessandro, Bonanni, Luca, Zenti, Maria Grazia, Matsuki, Kota, Hori, Mika, Ogura, Masatsune, Masuda, Daisaku, Kobayashi, Takuya, Nagahama, Kumiko, Al-Jarallah, Mohammed, Radovic, Mirjana, Lunegova, Olga, Bektasheva, Erkayim, Khodzhiboboev, Elyor, Erglis, Andrejs, Gilis, Dainus, Nesterovics, Georgijs, Saripo, Vita, Meiere, Ruta, Upena-RozeMicena, Arta, Terauda, Elizabete, Jambart, Selim, Khoury, Petra E., Elbitar, Sandy, Ayoub, Carine, Ghaleb, Youmna, Aliosaitiene, Urte, Kutkiene, Sandra, Kasim, Noor A.M., Nor, Noor S.M., Ramli, Anis S., Razak, Suraya A., Al-Khateeb, Alyaa, Kadir, Siti H.S.A., Muid, Suhaila A., Rahman, Thuhairah A., Kasim, Sazzli S., Radzi, Ahmad B.M., Ibrahim, Khairul S., Razali, Salmi, Ismail, Zaliha, Ghani, Rohana A., Hafidz, Muhammad I.A., Chua, Ang L., Rosli, Marshima M., Annamalai, Muthukkaruppan, Teh, Lay K., Razali, Rafezah, Chua, Yung A., Rosman, Azhari, Sanusi, Abdul R., Murad, Nor A.A., Jamal, A. Rahman A., Nazli, Sukma A., Razman, Aimi Z., Rosman, Norhidayah, Rahmat, Radzi, Hamzan, Nur S., Azzopardi, C., Mehta, Roopa, Martagon, Alexandro J., Ramirez, Gabriela A.G., Villa, Neftali E.A., Vazquez, Arsenio V., Elias-Lopez, Daniel, Retana, Gustavo G., Rodriguez, Betsabel, Macías, Jose J.C., Zazueta, Alejandro R., Alvarado, Rocio M., Portano, Julieta D.M., Lopez, Humberto A., Sauque-Reyna, Leobardo, Herrera, Laura G.G., Mendia, Luis E.S., Aguilar, Humberto Garcia, Cooremans, Elizabeth R., Aparicio, Berenice P., Zubieta, Victoria M., Gonzalez, Perla A.C., Ferreira-Hermosillo, Aldo, Portilla, Nacu C., Dominguez, Guadalupe J., Garcia, Alinna Y.R., Cazares, Hector E.A., Gonzalez, Jesus R., Valencia, Carla V.M., Padilla, Francisco G., Prado, Ramon M., De los Rios Ibarra, Manuel O., Villicaña, Ruy D.A., Rivera, Karina J.A., Carrera, Ricardo A., Alvarez, Jose A., Martinez, Jose C.A., de los Reyes Barrera Bustillo, Manuel, Vargas, Gonzalo C., Chacon, Roberto C., Andrade, Mario H.F., Ortega, Ashanty F., Alcala, Hector G., de Leon, Laura E.G., Guzman, Berenice G., Garcia, Jose J.G., Cuellar, Juan C.G., Cruz, Jose R.G., Garcia, Anell Hernandez, Almada, Jesus R.H., Herrera, Ursulo J., Sobrevilla, Fabiola L., Rodriguez, Eduardo M., Sibaja, Cristina M., Rodriguez, Alma B.M., Oyervides, Jose C.M., Vazquez, Daniel I.P., Rodriguez, Eduardo A.R., Osorio, Ma L.R., Saucedo, Juan R., Tamayo, Margarita T., Talavera, Luis A.V., Arroyo, Luis E.V., Carrillo, Eloy A.Z., Isara, Alphonsus, Obaseki, Darlington E., Al-Waili, Khalid, Al-Zadjali, Fahad, Al-Zakwani, Ibrahim, Al-Kindi, Mohammed, Al-Mukhaini, Suad, Al-Barwani, Hamida, Rana, Asim, Shah, Lahore S.U., Starostecka, Ewa, Konopka, Agnieszka, Lewek, Joanna, Bartłomiejczyk, Marcin, Gąsior, Mariusz, Dyrbuś, Krzysztof, Jóźwiak, Jacek, Gruchała, Marcin, Pajkowski, Marcin, Romanowska-Kocejko, Marzena, Żarczyńska-Buchowiecka, Marta, Chmara, Magdalena, Wasąg, Bartosz, Parczewska, Aleksandra, Gilis-Malinowska, Natasza, Borowiec-Wolna, Justyna, Stróżyk, Aneta, Woś, Marlena, Michalska-Grzonkowska, Aleksandra, Medeiros, Ana M., Alves, Ana C., Silva, Francisco, Lobarinhas, Goreti, Palma, Isabel, de Moura, Jose P., Rico, Miguel T., Rato, Quitéria, Pais, Patrícia, Correia, Susana, Moldovan, Oana, Virtuoso, Maria J., Salgado, Jose M., Colaço, Ines, Dumitrescu, Andreea, Lengher, Calin, Mosteoru, Svetlana, Meshkov, Alexey, Ershova, Alexandra, Rozkova, Tatiana, Korneva, Victoria, Yu, Kuznetsova T., Zafiraki, Vitaliy, Voevoda, Mikhail, Gurevich, Victor, Duplyakov, Dmitry, Ragino, Yulia, Safarova, Maya, Shaposhnik, Igor, Alkaf, Fahmi, Khudari, Alia, Rwaili, Nawal, Al-Allaf, Faisal, Alghamdi, Mohammad, Batais, Mohammed A., Almigbal, Turky H., Kinsara, Abdulhalim, AlQudaimi, Ashraf H.A., Awan, Zuhier, Elamin, Omer A., Altaradi, Hani, Rajkovic, Natasa, Popovic, Ljiljana, Singh, Sandra, Stosic, Ljubica, Rasulic, Iva, Lalic, Nebojsa M., Lam, Carolyn, Le, Tan J., Siang, Eric L.T., Dissanayake, Sanjaya, I-Shing, Justin T., Shyong, Tai E., Jin, Terrance C.S., Balinth, Karin, Buganova, Ingrid, Fabryova, Lubomira, Kadurova, Michaela, Klabnik, Alexander, Kozárová, Miriam, Sirotiakova, Jana, Battelino, Tadej, Kovac, Jernej, Mlinaric, Matej, Sustar, Ursa, Podkrajsek, Katarina T., Fras, Zlatko, Jug, Borut, Cevc, Matija, Pilcher, Gillian J., Blom, D.J., Wolmarans, K.H., Brice, B.C., Muñiz-Grijalvo, Ovidio, Díaz-Díaz, Jose L., de Isla, Leopoldo P., Fuentes, Francisco, Badimon, Lina, Martin, François, Lux, Angela, Chang, Nien-Tzu, Ganokroj, Poranee, Akbulut, Mehmet, Alici, Gökhan, Bayram, Fahri, Can, Levent H., Celik, Ahmet, Ceyhan, Ceyhun, Coskun, Fatma Y., Demir, Mesut, Demircan, Sabri, Dogan, Volkan, Durakoglugil, Emre, Dural, Ibrahim E., Gedikli, Omer, Hacioglu, Aysa, Ildizli, Muge, Kilic, Salih, Kirilmaz, Bahadir, Kutlu, Merih, Oguz, Aytekin, Ozdogan, Oner, Onrat, Ersel, Ozer, Savas, Sabuncu, Tevfik, Sahin, Tayfun, Sivri, Fatih, Sonmez, Alper, Temizhan, Ahmet, Topcu, Selim, Tuncez, Abdullah, Vural, Mirac, Yenercag, Mustafa, Yesilbursa, Dilek, Yigit, Zerrin, Yildirim, Aytul B., Yildirir, Aylin, Yilmaz, Mehmet B., Atallah, Bassam, Traina, Mahmoud, Sabbour, Hani, Hay, Dana A., Luqman, Neama, Elfatih, Abubaker, Abdulrasheed, Arshad, Kwok, See, Oca, Nicolas D., Reyes, Ximena, Alieva, Rano B., Kurbanov, Ravshanbek D., Hoshimov, Shavkat U., Nizamov, Ulugbek I., Ziyaeva, Adolat V., Abdullaeva, Guzal J., Do, Doan L., Nguyen, Mai N.T., Kim, Ngoc T., Le, Thanh T., Le, Hong A., Tokgozoglu, Lale, Catapano, Alberico L., and Ray, Kausik K.

Published
2021
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57. Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails

Author

Biskup, Marek, Koenig, Wolfgang, and Santos, Renato Soares dos

Subjects
Mathematics - Probability, Mathematical Physics, Mathematics - Analysis of PDEs, 60H25, 82B44
Abstract

We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=\Delta u+\xi u$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $\Delta$ is the discrete Laplacian on $\mathbb Z^d$ and $\xi=(\xi(z))_{z\in\mathbb Z^d}$ is an i.i.d.\ random field with doubly-exponential upper tails. We prove that, for large $t$ and with large probability, a majority of the total mass $U(t):=\sum_x u(x,t)$ of the solution resides in a bounded neighborhood of a site $Z_t$ that achieves an optimal compromise between the local Dirichlet eigenvalue of the Anderson Hamiltonian $\Delta+\xi$ and the distance to the origin. The processes $t\mapsto Z_t$ and $t \mapsto \tfrac1t \log U(t)$ are shown to converge in distribution under suitable scaling of space and time. Aging results for $Z_t$, as well as for the solution to the parabolic problem, are also established. The proof uses the characterization of eigenvalue order statistics for $\Delta+\xi$ in large sets recently proved by the first two authors., Comment: 69 pages, version to appear in Prob. Theory Rel. Fields

Published
2016
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58. Mean-Field interacton of Brownian occupation measures. II: A rigorous construction of the Pekar process

Author

Bolthausen, Erwin, Koenig, Wolfgang, and Mukherjee, Chiranjib

Subjects
Mathematics - Probability, Mathematical Physics
Abstract

We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [6] in terms of the {\it{Pekar variational formula}}, which coincides with the behavior of the partition function of the {\it{polaron problem}} under strong coupling. Based on this, in 1986 Spohn [14] made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the {\it{Pekar process}}, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the "mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [11], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [8], as well as an idea inspired by a {\it{partial path exchange}} argument appearing in [1]., Comment: Following referee's suggestion, Section 4 is expanded and Proof of Theorem 2.1 is corrected therein. To appear in "Comm. Pure. Appl. Math."

Published
2015

60. Meta-analysis uncovers genome-wide significant variants for rapid kidney function decline

Author

Alizadeh, Behrooz Z., Boezen, H. Marike, Franke, Lude, van der Harst, Pim, Navis, Gerjan, Rots, Marianne, Snieder, Harold, Swertz, Morris, Wolffenbuttel, Bruce H.R., Wijmenga, Cisca, Abecasis, Goncalo, Baras, Aris, Cantor, Michael, Coppola, Giovanni, Economides, Aris, Lotta, Luca A., Overton, John D., Reid, Jeffrey G., Shuldiner, Alan, Beechert, Christina, Forsythe, Caitlin, Fuller, Erin D., Gu, Zhenhua, Lattari, Michael, Lopez, Alexander, Schleicher, Thomas D., Padilla, Maria Sotiropoulos, Toledo, Karina, Widom, Louis, Wolf, Sarah E., Pradhan, Manasi, Manoochehri, Kia, Ulloa, Ricardo H., Bai, Xiaodong, Balasubramanian, Suganthi, Barnard, Leland, Blumenfeld, Andrew, Eom, Gisu, Habegger, Lukas, Hawes, Alicia, Khalid, Shareef, Maxwell, Evan K., Salerno, William, Staples, Jeffrey C., Jones, Marcus B., Mitnaul, Lyndon J., Gorski, Mathias, Jung, Bettina, Li, Yong, Matias-Garcia, Pamela R., Wuttke, Matthias, Coassin, Stefan, Thio, Chris H.L., Kleber, Marcus E., Winkler, Thomas W., Wanner, Veronika, Chai, Jin-Fang, Chu, Audrey Y., Cocca, Massimiliano, Feitosa, Mary F., Ghasemi, Sahar, Hoppmann, Anselm, Horn, Katrin, Li, Man, Nutile, Teresa, Scholz, Markus, Sieber, Karsten B., Teumer, Alexander, Tin, Adrienne, Wang, Judy, Tayo, Bamidele O., Ahluwalia, Tarunveer S., Almgren, Peter, Bakker, Stephan J.L., Banas, Bernhard, Bansal, Nisha, Biggs, Mary L., Boerwinkle, Eric, Bottinger, Erwin P., Brenner, Hermann, Carroll, Robert J., Chalmers, John, Chee, Miao-Li, Chee, Miao-Ling, Cheng, Ching-Yu, Coresh, Josef, de Borst, Martin H., Degenhardt, Frauke, Eckardt, Kai-Uwe, Endlich, Karlhans, Franke, Andre, Freitag-Wolf, Sandra, Gampawar, Piyush, Gansevoort, Ron T., Ghanbari, Mohsen, Gieger, Christian, Hamet, Pavel, Ho, Kevin, Hofer, Edith, Holleczek, Bernd, Xian Foo, Valencia Hui, Hutri-Kähönen, Nina, Hwang, Shih-Jen, Ikram, M. Arfan, Josyula, Navya Shilpa, Kähönen, Mika, Khor, Chiea-Chuen, Koenig, Wolfgang, Kramer, Holly, Krämer, Bernhard K., Kühnel, Brigitte, Lange, Leslie A., Lehtimäki, Terho, Lieb, Wolfgang, Loos, Ruth J.F., Lukas, Mary Ann, Lyytikäinen, Leo-Pekka, Meisinger, Christa, Meitinger, Thomas, Melander, Olle, Milaneschi, Yuri, Mishra, Pashupati P., Mononen, Nina, Mychaleckyj, Josyf C., Nadkarni, Girish N., Nauck, Matthias, Nikus, Kjell, Ning, Boting, Nolte, Ilja M., O’Donoghue, Michelle L., Orho-Melander, Marju, Pendergrass, Sarah A., Penninx, Brenda W.J.H., Preuss, Michael H., Psaty, Bruce M., Raffield, Laura M., Raitakari, Olli T., Rettig, Rainer, Rheinberger, Myriam, Rice, Kenneth M., Rosenkranz, Alexander R., Rossing, Peter, Rotter, Jerome I., Sabanayagam, Charumathi, Schmidt, Helena, Schmidt, Reinhold, Schöttker, Ben, Schulz, Christina-Alexandra, Sedaghat, Sanaz, Shaffer, Christian M., Strauch, Konstantin, Szymczak, Silke, Taylor, Kent D., Tremblay, Johanne, Chaker, Layal, van der Most, Peter J., Verweij, Niek, Völker, Uwe, Waldenberger, Melanie, Wallentin, Lars, Waterworth, Dawn M., White, Harvey D., Wilson, James G., Wong, Tien-Yin, Woodward, Mark, Yang, Qiong, Yasuda, Masayuki, Yerges-Armstrong, Laura M., Zhang, Yan, Wanner, Christoph, Böger, Carsten A., Köttgen, Anna, Kronenberg, Florian, Pattaro, Cristian, and Heid, Iris M.

Published
2021
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65. Mean-field interaction of Brownian occupation measures, I: uniform tube property of the Coulomb functional

Author

Koenig, Wolfgang and Mukherjee, Chiranjib

Subjects
Mathematics - Probability
Abstract

We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the present paper, we show that likewise the Coulomb functional of the occupation measure concentrates around the set of corresponding Coulomb functionals of the maximizers in the uniform topology. This is a decisive step on the way to a rigorous proof of the convergence of the normalized occupation measures towards an explicit mixture of the maximizers, derived in [BKM15]. Our methods rely on deriving H{\"o}lder-continuity of the Coulomb functional of the occupation measure with exponentially small deviation probabilities and invoking the large deviation theory developed in [MV14] to a certain shift-invariant functional of the occupation measures., Comment: To appear in: "Annales de l'Institut Henri Poincare"

Published
2015

72. Eigenvalue fluctuations for lattice Anderson Hamiltonians

Author

Biskup, Marek, Fukushima, Ryoki, and Koenig, Wolfgang

Subjects
Mathematics - Probability, Mathematical Physics, Mathematics - Spectral Theory, 60H25, 82B44, 35P20, 74Q15, 47A75, 47H40
Abstract

We study the statistics of Dirichlet eigenvalues of the random Schr\"odinger operator $-\epsilon^{-2}\Delta^{(\text{d})}+\xi^{(\epsilon)}(x)$, with $\Delta^{(\text{d})}$ the discrete Laplacian on $\mathbb Z^d$ and $\xi^{(\epsilon)}(x)$ uniformly bounded independent random variables, on sets of the form $D_\epsilon:=\{x\in \mathbb Z^d\colon x\epsilon\in D\}$ for $D\subset \mathbb R^d$ bounded, open and with a smooth boundary. If $\mathbb E\xi^{(\epsilon)}(x)=U(x\epsilon)$ holds for some bounded and continuous $U\colon D\to \mathbb R$, we show that, as $\epsilon\downarrow0$, the $k$-th eigenvalue converges to the $k$-th Dirichlet eigenvalue of the homogenized operator $-\Delta+U(x)$, where $\Delta$ is the continuum Dirichlet Laplacian on $D$. Assuming further that $\text{Var}(\xi^{(\epsilon)}(x))=V(x\epsilon)$ for some positive and continuous $V\colon D\to \mathbb R$, we establish a multivariate central limit theorem for simple eigenvalues centered by their expectation. The limiting covariance for a given pair of simple eigenvalues is expressed as an integral of $V$ against the product of squares of the corresponding eigenfunctions of $-\Delta+U(x)$., Comment: 26 pages, to appear in SIAM J. Math. Anal

Published
2014
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75. Eigenvalue order statistics for random Schr\'odinger operators with doubly-exponential tails

Author

Biskup, Marek and Koenig, Wolfgang

Subjects
Mathematics - Probability, Mathematical Physics, 60H25, 82B44, 82C44, 60J80
Abstract

We consider random Schr\"odinger operators of the form $\Delta+\xi$, where $\Delta$ is the lattice Laplacian on $\mathbb Z^d$ and $\xi$ is an i.i.d. random field, and study the extreme order statistics of the eigenvalues for this operator restricted to large but finite subsets of $\mathbb Z^d$. We show that for $\xi$ with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class. The corresponding eigenfunctions are exponentially localized in regions where $\xi$ takes large, and properly arranged, values. A new and self-contained argument is thus provided for Anderson localization at the spectral edge which permits a rather explicit description of the shape of the potential and the eigenfunctions. Our study serves as an input into the analysis of an associated parabolic Anderson problem., Comment: 36 pages

Published
2013
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76. Moment asymptotics for multitype branching random walks in random environment

Author

Gün, Onur, König, Wolfgang, and Sekulović, Ozren

Subjects
Mathematics - Probability, 60J80, 60J55, 60F10, 60K37, 60J10
Abstract

We study a discrete time multitype branching random walk on a finite space with finite set of types. Particles follow a Markov chain on the spatial space whereas offspring distributions are given by a random field that is fixed throughout the evolution of the particles. Our main interest lies in the averaged (annealed) expectation of the population size, and its long-time asymptotics. We first derive, for fixed time, a formula for the expected population size with fixed offspring distributions, which is reminiscent of a Feynman-Kac formula. We choose Weibull-type distributions with parameter $1/\rho_{ij}$ for the upper tail of the mean number of $j$ type particles produced by an $i$ type particle. We derive the first two terms of the long-time asymptotics, which are written as two coupled variational formulas, and interpret them in terms of the typical behavior of the system.

Published
2013

78. World Health Organization cardiovascular disease risk charts: revised models to estimate risk in 21 global regions

Author

Kaptoge, Stephen, Pennells, Lisa, De Bacquer, Dirk, Cooney, Marie Therese, Kavousi, Maryam, Stevens, Gretchen, Riley, Leanne Margaret, Savin, Stefan, Khan, Taskeen, Altay, Servet, Amouyel, Philippe, Assmann, Gerd, Bell, Steven, Ben-Shlomo, Yoav, Berkman, Lisa, Beulens, Joline W, Björkelund, Cecilia, Blaha, Michael, Blazer, Dan G, Bolton, Thomas, Bonita Beaglehole, Ruth, Brenner, Hermann, Brunner, Eric J, Casiglia, Edoardo, Chamnan, Parinya, Choi, Yeun-Hyang, Chowdry, Rajiv, Coady, Sean, Crespo, Carlos J, Cushman, Mary, Dagenais, Gilles R, D'Agostino Sr, Ralph B, Daimon, Makoto, Davidson, Karina W, Engström, Gunnar, Ford, Ian, Gallacher, John, Gansevoort, Ron T, Gaziano, Thomas Andrew, Giampaoli, Simona, Grandits, Greg, Grimsgaard, Sameline, Grobbee, Diederick E, Gudnason, Vilmundur, Guo, Qi, Tolonen, Hanna, Humphries, Steve, Iso, Hiroyasu, Jukema, J Wouter, Kauhanen, Jussi, Kengne, Andre Pascal, Khalili, Davood, Koenig, Wolfgang, Kromhout, Daan, Krumholz, Harlan, Lam, TH, Laughlin, Gail, Marín Ibañez, Alejandro, Meade, Tom W, Moons, Karel G M, Nietert, Paul J, Ninomiya, Toshiharu, Nordestgaard, Børge G, O'Donnell, Christopher, Palmieri, Luigi, Patel, Anushka, Perel, Pablo, Price, Jackie F, Providencia, Rui, Ridker, Paul M, Rodriguez, Beatriz, Rosengren, Annika, Roussel, Ronan, Sakurai, Masaru, Salomaa, Veikko, Sato, Shinichi, Schöttker, Ben, Shara, Nawar, Shaw, Jonathan E, Shin, Hee-Choon, Simons, Leon A, Sofianopoulou, Eleni, Sundström, Johan, Völzke, Henry, Wallace, Robert B, Wareham, Nicholas J, Willeit, Peter, Wood, David, Wood, Angela, Zhao, Dong, Woodward, Mark, Danaei, Goodarz, Roth, Gregory, Mendis, Shanthi, Onuma, Oyere, Varghese, Cherian, Ezzati, Majid, Graham, Ian, Jackson, Rod, Danesh, John, and Di Angelantonio, Emanuele

Published
2019
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79. Integrierte Anwendungssysteme im Unternehmen

Author

Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, Buxmann, Peter, Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, and Buxmann, Peter

Published
2017
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80. Management der Ressource IT

Author

Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, Buxmann, Peter, Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, and Buxmann, Peter

Published
2017
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81. Digitale Transformation von Unternehmen

Author

Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, Buxmann, Peter, Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, and Buxmann, Peter

Published
2017
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82. Rechner und deren Vernetzung

Author

Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, Buxmann, Peter, Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, and Buxmann, Peter

Published
2017
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84. Daten, Informationen und Wissen

Author

Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, Buxmann, Peter, Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, and Buxmann, Peter

Published
2017
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85. Grundlagen

Author

Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, Buxmann, Peter, Mertens, Peter, Bodendorf, Freimut, König, Wolfgang, Schumann, Matthias, Hess, Thomas, and Buxmann, Peter

Published
2017
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86. Large deviations for the local times of a random walk among random conductances in a growing box

Author

König, Wolfgang and Wolff, Tilman

Subjects
Mathematics - Probability, 60K37, 60J65, 60J55, 60F10
Abstract

We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in $\Z^d$. We work in the interesting case that the conductances are positive, but may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small conductance values and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution as well as the time-dependent size of the box. An interesting phase transition occurs if the thickness parameter of the conductance tails exceeds a certain threshold: for thicker tails, the random walk spreads out over the entire growing box, for thinner tails it stays confined to some bounded region. In fact, in the first case, the rate function turns out to be equal to the $p$-th power of the $p$-norm of the gradient of the square root for some $p\in(\frac {2d}{d+2},2)$. This extends the Donsker-Varadhan-G\"artner rate function for the local times of Brownian motion (with deterministic environment) from $p=2$ to these values. As corollaries of our LDP, we derive the logarithmic asymptotics of the non-exit probability of the RWRC from the growing box, and the Lifshitz tails of the generator of the RWRC, the randomised Laplace operator. To contrast with the annealed, not uniformly elliptic case, we also provide an LDP in the quenched setting for conductances that are bounded and bounded away from zero. The main tool here is a spectral homogenisation result, based on a quenched invariance principle for the RWRC., Comment: 32 pages

Published
2013

87. Connection times in large ad-hoc mobile networks

Author

Döring, Hanna, Faraud, Gabriel, and König, Wolfgang

Subjects
Mathematics - Probability
Abstract

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, that is, they are iteratively forwarded from participant to participant over distances smaller than the communication radius until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model. Here, we give a negative upper bound for the decay rate, in the limit of large time horizons, of the probability of the event that the portion of the connection time is less than the expectation., Comment: Published at http://dx.doi.org/10.3150/15-BEJ724 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Published
2013
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89. Residual vascular risk in diabetes – Will the SPPARM alpha concept hold the key?

Author

Fruchart, Jean-Charles, Santos, Raul D., Yamashita, Shizuya, Aikawa, Masanori, Libby, Peter, Plutzky, Jorge, Pradhan, Aruna, Ridker, Paul, Aguilar-Salinas, Carlos, Al Rasadi, Khalid, Amarenco, Pierre, Barter, Philip J., Ceska, Richard, Corsini, Alberto, Ruscica, Massimiliano, Després, Jean-Pierre, Duriez, Patrick, Eckel, Robert H., Ezhov, Marat V., Farnier, Michel, Ginsberg, Henry N., Hermans, Michel P., Ishibashi, Shun, Karpe, Fredrik, Kodama, Tatsuhiko, Koenig, Wolfgang, Krempf, Michel, Lim, Soo, Lorenzatti, Alberto J., McPherson, Ruth, Nuñez-Cortes, Jesus Millan, Nordestgaard, Børge G., Ogawa, Hisao, Packard, Chris J., Ponte-Negretti, Carlos I., Reiner, Željko, Sadikot, Shaukat, Shimano, Hitoshi, Sritara, Piyamitr, Stock, Jane K., Su, Ta-Chen, Susekov, Andrey V., Tartar, André, Taskinen, Marja-Riitta, Tenenbaum, Alexander, Tokgözoğlu, Lale S., Tomlinson, Brian, Tybjærg-Hansen, Anne, Valensi, Paul, Vrablík, Michal, Wahli, Walter, Watts, Gerald F., Yokote, Koutaro, and Zambon, Alberto

Published
2019
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91. Moment asymptotics for branching random walks in random environment

Author

Gün, Onur, König, Wolfgang, and Sekulović, Ozren

Subjects
Mathematics - Probability, 60J80, 60J55, 60F10, 60K37
Abstract

We consider the long-time behaviour of a branching random walk in random environment on the lattice $\Z^d$. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of spatially dependent killing/branching rates. The main objects of our interest are the annealed moments $< m_n^p > $, i.e., the $p$-th moments over the medium of the $n$-th moment over the migration and killing/branching, of the local and global population sizes. For $n=1$, this is well-understood \cite{GM98}, as $m_1$ is closely connected with the parabolic Anderson model. For some special distributions, \cite{A00} extended this to $n\geq2$, but only as to the first term of the asymptotics, using (a recursive version of) a Feynman-Kac formula for $m_n$. In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that $< m_n^p >$ and $< m_1^{np} >$ are asymptotically equal, up to an error $\e^{o(t)}$. The cornerstone of our method is a direct Feynman-Kac-type formula for $m_n$, which we establish using the spine techniques developed in \cite{HR11}., Comment: 18 pages, 3 figures

Published
2012

94. Large deviations for cluster size distributions in a continuous classical many-body system

Author

Jansen, Sabine, König, Wolfgang, and Metzger, Bernd

Subjects
Mathematics - Probability, Mathematical Physics
Abstract

An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $\beta\in (0,\infty)$ and particle density $\rho\in(0,\rho_{\mathrm{cp}})$ in the thermodynamic limit. Here $\rho_{\mathrm{cp}}>0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $\Gamma$-convergence of the rate function toward an explicit limiting rate function in the low-temperature dilute limit $\beta\to \infty$, $\rho\downarrow0$ such that $-\beta^{-1}\log\rho\to\nu$ for some $\nu\in(0,\infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the decoupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $\nu$. Under additional assumptions on the potential, the $\Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle., Comment: Published in at http://dx.doi.org/10.1214/14-AAP1014 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2011
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96. Large deviations for the local times of a random walk among random conductances

Author

König, Wolfgang, Salvi, Michele, and Wolff, Tilman

Subjects
Mathematics - Probability, 60F10 (Primary), 60G50, 60J65, 60J55 (Secondary)
Abstract

We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $\Z^d$ in the spirit of Donsker-Varadhan \cite{DV75}. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain., Comment: 12 pages

Published
2011

97. The longest excursion of a random interacting polymer

Author

Köcher, Janine and König, Wolfgang

Subjects
Mathematics - Probability, 60F05, 82D60
Abstract

We consider a random $N$-step polymer under the influence of an attractive interaction with the origin and derive a limit law -- after suitable shifting and norming -- for the length of the longest excursion towards the Gumbel distribution. The embodied law of large numbers in particular implies that the longest excursion is of order $\log N$ long. The main tools are taken from extreme value theory and renewal theory., Comment: 5 pages

Published
2011

98. A two cities theorem for the parabolic Anderson model

Author

König, Wolfgang, Lacoin, Hubert, Mörters, Peter, and Sidorova, Nadia

Subjects
Mathematics - Probability
Abstract

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_tu(t,z)=\Delta u(t,z)+\xi(z)u(t,z)$ on $(0,\infty)\times {\mathbb{Z}}^d$ with random potential $(\xi(z):z\in{\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $\xi(z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem., Comment: Published in at http://dx.doi.org/10.1214/08-AOP405 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Published
2011
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99. Upper tails of self-intersection local times of random walks: survey of proof techniques

Author

König, Wolfgang

Subjects
Mathematics - Probability, 60K37, 60F10, 60J55
Abstract

The asymptotics of the probability that the self-intersection local time of a random walk on $\Z^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated techniques and results. This is an extended summary of a talk held on the CIRM-conference on {\it Excess self-intersection local times, and related topics} in Luminy, 6-10 Dec., 2010., Comment: 11 pages

Published
2010

100. Brownian motion in a truncated Weyl chamber

Author

König, Wolfgang and Schmid, Patrick

Subjects
Mathematics - Probability, 60J65, 60F10
Abstract

We examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.

Published
2010

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