Christian Borgs

Picture of Christian BorgsChristian Borgs is a professor of Computer Science in the EECS department at the University of California, Berkeley and a member of the Berkeley Artificial Intelligence Research (BAIR) Lab.

Christian Borgs graduated in Physics at the University of Munich, and holds a Ph.D. in Mathematical Physics from the the University of Munich and the Max-Planck-Institute for Physics, as well as a Habilitation in Mathematical Physics from the Free University in Berlin. He was the C4 chair of Statistical Mechanics at the University of Leipzig, before joining Microsoft Reseach in 1997 to co-found the Theory Group. He was manager of the Theory Group until 2008, when he co-founded Microsoft Research New England in Cambridge, Massachusetts, a lab that brings together traditional CS and statistics research with economics as well as qualitative social science research. Borgs was Deputy Managing Director of this lab from 2008 until he left Microsoft for Berkeley in 2020.

Among the honors he has received are the Karl-Scheel Prize of the German Physical Society, and the Heisenberg Fellowship of the German Research Council. Borgs has twice been a member of the Institute for Advanced Study in Princeton. He is a a Fellow of the American Mathematical Society, and the Association of the Advancement of Science. He served or serves on several editorial boards, prize committees, and governing boards, including the Board of Trustees of the Institute for Pure and Applied Mathematics (IPAM), the Governing Board of the Institute for Mathematics and its Applications (IMA) where he was chair from 2017 until 2019, the Adivosry Board of the Harvard Insitute for Applied Computer Science (IACS) and the External Advisory Board for Computing at NYU.

Christian Borgs has authored almost 140 research papers and is named as coinventor of over 30 patents. His current research focuses on the science of networks, including mathematical foundations, particularly the theory of graph limits (which he co-invented about 15 years ago), graph processes, graph algorithms, and applications of graph theory from economics to systems biology. Borgs is also well known for his earlier work on mathematical statistical physics, including the theory of first-order phase transitions and finite-size effects. He was one of the first to apply methods from mathematical statistical physics to problems in theoretical computer science, including phase transitions in combinatorial optimization, and the study of Markov chains. He has recently begun to work on aspects of responsible AI, from differential privacy to questions of bias in automatic decision making.

Christian Borgs is married to Jennifer Chayes, who also works at Bekeley, and with whom he collaborates on most of his scientific work. In his spare time, he enjoys art, theatre and classical music, as well as skiing and swimming.

Selected Publications (see here for the complete list)

Graph Limits, Graphons and Nonparametric Network Models

Bolyai Society Mathematical Studies, Vol. 28, 29–157 (2020).
PDF Counting graph homomorphisms (with J. T. Chayes, L. Lovasz, V. Sos, B. Szegedy and K. Vesztergombi). In Topics in Discrete Mathematics (eds. M. Klazar, J. Kratochvil, M. Loebl, J. Matousek, R. Thomas, P. Valtr), 315-371, Springer (2006).
PDF Graph limits and parameter testing (with J. T. Chayes, L. Lovasz, V. Sos, B. Szegedy and K. Vesztergombi). Proceedings of the 38rd Annual ACM Symposium on the Theory of Computing (STOC), 261-270 (2006).
PDF Convergent sequences of dense graphs I: Subgraph frequencies, metric properties and testing (with J. T. Chayes, L. Lovasz, V. Sos, and K. Vesztergombi). Advances in Math. 219, 1801-1851 (2008).
PDF Convergent sequences of dense graphs II: Multiway cuts and statistical physics (with J. T. Chayes, L. Lovasz, V. Sos, and K. Vesztergombi).  Ann. of Math. 176, 151-219 (2012).
PDF Moments of two-variable functions and the uniqueness of graph limits (with J. T. Chayes and L. Lovasz). GAFA 19, 1597-1619 (2010).
PDF Limits of randomly grown graph sequences (with J. T. Chayes, L. Lovasz, V. Sos, K. Veszterbombi). Eur. J. Comb. 32, 985-999 (2011).
PDF Left and right convergence of graphs with bounded degree (with J. T. Chayes, J. Kahn, L. Lovasz). Random Struct. Alg. 42, 1-28 (2013).
PDF Asymptotic behavior and distributional limits of preferential attachment graphs (with N. Berger, J. T. Chayes, A. Saberi). Ann. Probab. 42(1), 1-40 (2014).
PDF Convergent sequences of sparse graphs: A large deviations approach (with J. T. Chayes and D. Gamarnik). Random Struct. Alg. 51, 52-89 (2017)
PDF An Lp theory of sparse graph convergence I: limits, sparse random graph models, and power law distributions (with J. T. Chayes, H. Cohn and Y. Zhao). Trans. Amer. Math. Soc.  372, 3019-3062 (2019).
PDF An Lp theory of sparse graph convergence II: LD convergence, quotients, and right convergence (with J. T. Chayes, H. Cohn and Y. Zhao). Annals of Prob. 45, 337-396 (2018).
PDF Graphons: A nonparametric method to model, estimate, and design algorithms for massive networks (with J. T. Chayes). Proceedings of the 18th ACM Conference on Economics and Computation (EC’17), 665–672, (2017).
PDF Sparse exchangeable graphs and their limits via graphon processes (with J. T. Chayes, H. Cohn and N. Holden). Journal of Machine Learning Research 18 (210), 1-71 (2018).
PDF Sampling perspectives on sparse exchangeable graphs (with J.T. Chayes, H. Cohn, V. Veitch). Annals of Prob. 47 (5), 2754-2800 (2019).
PDF Identifiability for graphexes and the weak kernel metric (with J.T. Chayes, H. Cohn, L. M. Lovasz). In: I. Brny, G. Katona, A. Sali, Attila (Eds.),
Building Bridges II - Mathematics of Laszlo Lovasz. Bolyai Society Mathematical Studies, Vol. 28, 29–157 (2020).
PDF Limits of sparse conguration models and beyond: graphexes and multi-graphexes (with J.T. Chayes, S. Dhara, and S. Sen). Preprint (2019).
PDF A correction to Kallenberg's theorem for jointly exchangeable random measures (with J.T. Chayes, S. Dhara, and S. Sen). Preprint (2019).
PDF A large deviation principle for block models (with J.T. Chayes, J. Gaudio, S. Petti and S. Sen). Preprint (2020).

Mechanism Design, Social Choice and Recommendation Systems

PDF Multi-unit auctions with budget-constrained bidders (with J. T. Chayes, N. Immorlica, M. Mahdian and A. Saberi). Proceedings of the 6th ACM Conference on Electronic Commerce (EC), 44-51 (2005).
PDF Bid optimization in online advertisement auctions (with J. T. Chayes, O. Etesami, N. Immorlica and M. Mahdian). 2nd Workshop on Sponsored Search Auctions (2006) and Proceedings of the 16th international conference on World Wide Web (WWW), 531-540 (2007).
PDF The myth of the folk theorem (with J. T. Chayes, N. Immorlica, A. Kalai, V. Mirrokni and C. Papadimitriou). Proceedings of the 40st Annual ACM Symposium on the Theory of Computing (STOC), 365-372 (2008).
PDF Trust-based recommendation systems: An axiomatic approach (with R. Andersen, J. T. Chayes, U.Feige, A. Flaxman, A. Kalai, V. Mirrokni and M. Tennenholtz). Proceedings of the 17th international conference on World Wide Web (WWW), 199-208 (2008).
PDF A novel approach to propagating distrust (with J. T. Chayes, A. Kalai, A. Malekiany, M. Tennenholtz). Proceedings of the 6th International Workshop on Internet and Network Economics (WINE) 87-105 (2010).
PDF Game-theoretic models of information overload in social networks (with J. T. Chayes, B. Karrer, B. Meeder, R. Ravi, R. Reagans and A. Sayedi). Proceedings of the 7th Workshop on Algorithms and Models for the Web Graph (WAW) 146 - 161 (2010).
PDF Fast convergence of natural bargaining dynamics in exchange networks (with Y. Kanoria, M. Bayati, J. T. Chayes, and A. Montanari). Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithm (SODA), 1518-1537 (2011).
PDF The hitchhiker's guide to affiliation networks: A game-theoretic approach (with J. T. Chayes, J. Ding and B. Lucier). Proceedings of the 2nd Symposium on Innovations in Computer Science (ICS), 389-400 (2011).
PDF Pricing and queuing (with J.T. Chayes, S. Doroudi, M. Harchol-Balter, K. Xu). ACM SIGMETRICS Performance Evaluation Review 40(3): 71-73 (2012).
PDF Priority pricing in queues with a continuous distribution of customer valuations (S. Doroudi, M. Akan, M. Harchol-Balter, J. Karp, C. Borgs, J.T. Chayes). CMU technical report CMU-CS-13-109 (2013).
PDF The optimal admission threshold in observable queues with state dependent pricing (with J.T. Chayes, S. Doroudi, M. Harchol-Balter, K. Xu). Probability in the Engineering and Informational Sciences 28, 101-110 (2014).
PDF Optimal multi-period pricing with service guarantees (with O. Candogan, J.T. Chayes, I. Lobel, and H. Nazerzadeh). Management Science 60, 1792-1811 (2014).
PDF Bargaining dynamics in exchange networks (with M. Bayati, J.T. Chayes, Y. Kanoria and A. Montanari). J. Econ. Theory 156, 417-454 (2015)
PDF An axiomatic approach to community detection (with C. Borgs, A. Marple, and S.-H. Teng). Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science (ITCS '16), 135-146 (2016).
PDF Thy friend is my friend: iterative collaborative filtering for sparse matrix estimation (with J.T. Chayes, C.E. Lee, D.Shah). Advances in Neural Information Processing Systems (NIPS) 30, 4718–4729, (2017).
PDF Iterative collaborative filtering for sparse matrix estimation (with J.T. Chayes, C.E. Lee, D.Shah). Preprint (2017).

Machine Learning, Estimation and Responsible AI

PDF Statistical mechanics of Steiner trees (with M. Bayati, A. Braunstein, J. Chayes, A. Ramezanpour, and R. Zecchina). Physical Review Letters 101, 037208 (2008), reprinted in Virtual Journal of Biological Physics Research 16, August 1 (2008).
PDF Belief-Propagation for weighted b-matchings on arbitrary graphs and its relation to linear programs with integer solutions (with M. Bayati, J. Chayes and R. Zecchina) SIAM Journal of Discrete Mathematics, 25, 989-1011 (2011).
PDF Private graphon estimation for sparse graphs (with J. T. Chayes and A. Smith). Advances in Neural Information Processing Systems (NIPS) 28, 1369--1377 (2015).
PDF Full version of the above paper.
PDF Consistent nonparametric estimation for heavy-tailed sparse graphs (with J. T. Chayes, H. Cohn and S. Ganguly). Preprint (2015).
PDF Unreasonable effectiveness of learning neural networks: From accessible states and robust ensembles to basic algorithmic schemes (with C. Baldassi, J. T. Chayes, A. Ingrosso, C. Lucibello, L. Saglietti, and R. Zecchina). Proceedings of the National Academy of Sciences (PNAS) 113, E7655–E7662 (2016).
PDF Entropy-SGD: biasing gradient descent into wide valleys (P. Chaudhari, A. Choromanska, S. Soatto, Y. LeCun, C. Baldassi, C. Borgs, J.T. Chayes, L. Sagun and R. Zecchina). International Conference on Learning Representations (ICLR), 1-19 (2017).
PDF Revealing network structure, condentially: improved rates for node private graphon estimation (with J.T. Chayes, A. Smith, I. Zadik). Proceedings of the 59 th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 533-543 (2018).
PDF Bias in bios: a case study of semantic representation bias in a high-stakes setting (M. De-Arteaga, A. Romanov, H. Wallach, J. Chayes, C. Borgs, A. Chouldechova, S. Geyik, K. Kenthapadi, A. Kalai). Proceedings of the 2nd Conference on Fairness, Accountability, and Transparency (FAT* '19), 120-128 (2019).
PDF What’s in a name? Reducing bias in bios without access to protected attributes. (A. Romanov, M. De-Arteaga, H. Wallach, J. Chayes, C. Borgs, A. Chouldechova, S. Geyik, K. Kenthapadi, A. Kalai). Proceedings of the 2019 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL '19). Best paper award.
PDF Algorithmic greenlining: an approach to increase diversity (with J. Chayes, N. Haghtalab, A. Kalai and E. Vitercik). AIII / ACM Conference on Artificial Intelligence, Ethics and Society (AIES) 69–76, (2019).
PDF The disparate equilibria of algorithmic decision-making when individuals invest rationally (L. Liu, A. Wilson, N. Haghtalab, A. Kalai, C. Borgs, and J.T. Chayes). Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency (FAT* '20), 381–391 (2020).

Computational Biology

PDF Finding undetected protein associations in cell signaling by belief propagation (with M. Bailly-Bechet, A. Braunstein, J. Chayes, A.Dagkessamanskaia, J. Francois, and R. Zecchina). Proceedings of the National Academy of Sciences (PNAS) 108, 882-887 (2011).
PDF Simultaneous reconstruction of multiple signaling pathways via the prize-collecting Steiner forest problem (N.Tuncbag, A.Braunstein, A.Pagnani, S.C.Huang, J.Chayes, C.Borgs, R.Zecchina, E.Fraenkel). 16th Annual International Conference on Research in Computational Molecular Biology (RECOMB), 287 - 301 (2012) and Journal of Computational Biology 20, 124-136 (2013).
PDF Sharing information to reconstruct patient-specific pathways in heterogeneous diseases (A. Gitter, A. Braunstein, A. Pagnani, C. Baldassi, C. Borgs, J. Chayes, R. Zecchina, E. Fraenkel). Proceedings of the Pacific Symposium on Biocomputing (PSB), 39-50 (2014).
PDF A multifactorial model of T cell expansion and durable clinical benefit in response to a PD-L1 inhibitor (M. Leiserson, V. Syrgkanis, A. Gilson, M. Dudik, S. Gillett, J. Chayes, C. Borgs, D.F. Bajorin, J. Rosenberg, S. Funt, A. Snyder, L. Mackey). PLoS ONE 13(12): e0208422, (2018).

Processes on Networks and Graph Algorithms

PDF On the spread of viruses on the Internet (with N. Berger, J. T. Chayes and A. Saberi). Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithm (SODA), 301-310 (2005).
PDF Local computation of pagerank contributions (with R. Andersen, , J. T. Chayes., J. Hopcroft, V. Mirrokni and S. Teng). Proceedings of the  5th Workshop on Algorithms and Models for the Web Graph (WAW), 150-165 (2007).
PDF Robust PageRank and locally computable spam detection features (with R. Andersen, J. T. Chayes, J. E. Hopcroft, K. Jain, V.S. Mirrokni and S.H. Teng). AIRWeb 2008, 69-76 (2008).
PDF On the stability of web crawling and web search (with R. Andersen, J. T. Chayes, J. E. Hopcroft, V.S. Mirrokni and S.H. Teng). ISAAC 2008, 680-691 (2008).
PDF How to distribute antidote to control epidemics (with J. T. Chayes, A. Ganesh, and A. Saberi). Random Struct. Algorithms 37, 204-222 (2010).
PDF We know who you followed last summer: inferring social link creation times in twitter (with B. Meeder, B. Karrer, A. Sayedi, R. Ravi and J.T. Chayes). Proceedings of the 20th International World Wide Web Conference (WWW), 517-526 (2011).
PDF Belief-Propagation for weighted b-matchings on arbitrary graphs and its relation to linear programs with integer solutions (with M. Bayati, J. Chayes and R. Zecchina). SIAM Journal of Discrete Mathematics 25, 989-1011 (2011).
PDF A sublinear time algorithm for PageRank computations (with M. Brautbar, J.T. Chayes and S.-H. Teng). Proceedings of the 9th Workshop on Algorithms and Models for the Web Graph (WAW), 41-53 (2012).
PDF The power of local information in social networks (with M. Brautbar, J. Chayes, S. Khanna, B. Lucier). Proceedings of the 8th International Workshop on Internet and Network Economics (WINE), 406 - 419 (2012).
PDF Finding endogenously formed communities (with M.-F. Balcan, M. Braverman, J.T. Chayes and S.-H. Teng). 24th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA), 767-783 (2013).
PDF Multi-scale matrix sampling and sublinear-time PageRank (with M. Brautbar, J.T. Chayes and S.-H. Teng). Internet Mathematics 10, 20-48 (2014).
PDF Maximizing social influence in nearly optimal time (with M. Brautbar, J. Chayes and B. Lucier). Proceedings of the 25nd Annual ACM-SIAM Symposium on Discrete Algorithm (SODA), 946-957 (2014).

Network Modeling and Graph Theory

PDF Directed scale-free graphs (with B. Bollobas, J. T. Chayes and O. Riordan). Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 132-139 (2003).
PDF Degree distribution of the FKP network model (with N. Berger, B. Bollobas, J. T. Chayes and O. Riordan). Proceedings of the 30th International Colloquium on Automata, Languages and Programming (ICALP), 725-738, Lecture Notes in Computer Science 2719 (2003).
PDF Exploring the community structure of newsgroups (with J. T. Chayes, M. Mahdian and A. Saberi). Proceedings of the 10th ACM SIGKDD International Conference on Knowledge, Discovery and Data Mining (KKD), 783-787 (2004).
DATA Newsgroup cluster data referred to in the above paper.
PDF Competition-induced preferential attachment (with N. Berger, J. T. Chayes, R. D'Souza and R. D. Kleinberg). Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP), 208-221, Lecture Notes in Computer Science 3142 (2004).
PDF Degree distribution of competition-induced preferential attachment graphs (with N. Berger, J. T. Chayes, R. D'Souza and R. D. Kleinberg). Combinatorics,  Probability and Computing 14, 697-721 (2005).
PDF Emergence of tempered preferential attachment from optimization (with N. Berger, J. T. Chayes, R. D'Souza and R. D. Kleinberg). Proceedings of the National Academy of Sciences (PNAS) 104, 6112-6117 (2007), cover article.
PDF Fitting the WHOIS Internet data A short note with technical details left out in the above paper.
PDF First to market is not everything: an analysis of preferential attachment with fitness (with J. T. Chayes, C. Daskalakis and S.Roch). Proceedings of the 39rd annual ACM Symposium on the Theory of Computing (STOC), 135-144 (2007).

Phase Transitions in Combinatorics and Computer Science

PDF Uniform boundedness of crossing probabilities implies hyperscaling (with J. T. Chayes, H. Kesten and J. Spencer). Random Structures and Algorithms 15, 368-413 (1999).
PDF The birth of the infinite cluster: Finite-size scaling in percolation (with J. T. Chayes, H. Kesten and J. Spencer). Communications in Mathematical Physics 224, 153-204 (2001).
PDF The scaling window of the 2-SAT transition (with B. Bollobas, J. T. Chayes, J.H. Kim and D.B. Wilson). Random Structures and Algorithms 18, 201-256 (2001).
PDF Sharp threshold and scaling window for the integer partitioning problem (with J. T. Chayes and B. Pittel). Proceedings of the 33rd annual ACM Symposium on the Theory of Computing (STOC), 330-336 (2001).
PDF Phase transition and finite-size scaling for the integer partitioning problem (with J. T. Chayes and B. Pittel). Random Structures and Algorithms 19, 247-288 (2001).
PDF Constrained integer partitions (with J. T. Chayes, S. Mertens and B. Pittel). Proceedings of the 6th Latin American Symposium on Theoretical Information (LATIN), 59-68, Lecture Notes in Computer Science 2976 (2004).
PDF Phase diagram for the constrained integer partitioning problem (with J. T. Chayes, S. Mertens and B. Pittel). Random Structures and Algorithms 24, 315-380 (2004).
PDF Random subgraphs of finite graphs: I. The scaling window under the triangle condition (with J. T. Chayes, R. van der Hofstad, G. Slade and J. Spencer). Random Structures and Algorithms 27, 137-184 (2005).
PDF Random subgraphs of finite graphs: II. The lace expansion and the triangle condition (with J. T. Chayes, R. van der Hofstad, G. Slade and J. Spencer). Annals of Probability 33, 1886-1944 (2005).
PDF Random subgraphs of finite graphs: III. The phase transition for the n-cube (with J. T. Chayes, R. van der Hofstad, G. Slade and J. Spencer). Combinatorica 26, 395-410 (2006).
PDF The Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels (with J. T. Chayes, E. Mossel and S. Roch). 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 518-530 (2006).
PDF Proof of the local REM conjecture for number partitioning I: Constant energy scales (with J. T. Chayes, S. Mertens and C. Nair). Random Struct. Algorithms 34, 217-240 (2009).
PDF Proof of the local REM conjecture for number partitioning II: Growing energy scales (with J. T. Chayes, S. Mertens and C. Nair). Random Struct. Algorithms 34, 241-284 (2009).
PDF Percolation on dense graph sequences (with B. Bollobas, J. T. Chayes and O. Riordan). Ann. Probab. 38, 150-183 (2010).

Monte Carlo Markov Chains and Approximation Algorithms

PDF Torpid mixing of some Monte Carlo Markov Chain algorithms in statistical physics (with J. T. Chayes, A. Frieze, J.H. Kim, P. Tetali, E. Vigoda and V. Vu). Proceedings of the 40th Annual Symposium on Foundations of Computer Science (FOCS), 218-229 (1999).
PDF On the sampling problem for H-colorings on the hypercubic lattice (with J. T. Chayes, M. Dyer and P. Tetali) in Graphs, Morphisms and Statistical Physics (eds. J Nesetril and P Winkler).  DIMACS Series in Discrete Mathematics and Theoretical Computer Science 63, 13-28, American Mathematical Society (2004).
PDF Tight bounds for mixing of the Swendsen-Wang algorithm at the Potts transition point (with J. T. Chayes and P. Tetali). Probab. Theory Relat. Fields 152, 509 - 557 (2012).
PDF Efficient sampling and counting algorithms for the Potts model on Zd at all temperatures (with J. T. Chayes, T. Helmuth, W. Perkins and P. Tetali). STOC 2020, to appear.

Zeros of Complex Partition Functions and Lee-Yang Theory

PDF Gibbs states of graphical representations of the Potts model with external fields (with M. Biskup, J. T. Chayes and R. Kotecky). Journal of Mathematical Physics 41, 1170-1210 (2000).
PDF General theory of Lee-Yang zeros in models with first-order phase transitions (with M. Biskup, J. T. Chayes, L. Kleinwaks and R. Kotecky). Physical Review Letters 84, 4794-4797 (2000).
PDF Partition function zeros at first-order phase transitions: A general analysis (with M. Biskup, J. T. Chayes, L. Kleinwaks and R. Kotecky). Communications in Mathematical Physics 251, 79-131 (2004).
PDF Partition function zeros at first-order phase transitions: Piorogov-Sinai theory (with M. Biskup, J. T. Chayes and R. Kotecky). Journal of Statistical Physics 116, 97-155 (2004).
PDF Absence of zeros for the chromatic polynomial on bounded degree graphs Combinatorics,  Probability and Computing 15, 63-74 (2006).

Some Earlier Work in Statistical Physics and Lattice Gauge Theory

PDF Lattice Yang-Mills theory at nonzero temperatures and the confinement problem (with E. Seiler). Commun. Math. Phys. 91, 329-380 (1983).
PDF Confinement, deconfinement and freezing in lattice Yang - Mills theories with continuous time Commun. Math. Phys. 116, 309-342 (1988).
PDF A unified approach to phase diagrams in field theory and statistical mechanics (with J. Imbrie). Commun. Math. Phys. 128, 305-328 (1988).
PDF A rigorous theory of finite-size scaling at first-order phase transitions (with R. Kotecky). J. Stat. Phys. 61, 79 - 110 (1990).
PDF Finite-size scaling for Potts models (with R. Kotecky and S. Miracle-Sole). J. Stat. Phys. 62, 529 - 551 (1991).
PDF A new method to determine first-order transition points from finite-size data (with W. Janke). Phys. Rev. Lett. 68, 1738 - 1741 (1992).
PDF Finite-size scaling and surface tension from effective one-dimensional systems (with J. Imbrie). Commun. Math. Phys. 145, 235-280 (1992)
PDF Finite-size scaling for Potts models in long cylinders. Nucl. Phys. B384, 605-645 (1992).
PDF Crossover-finite-size scaling at first order transitions (with J. Imbrie). J. Stat. Phys. 69, 487-537 (1992).
PDF Low temperature phase diagrams for quantum perturbations of classical spin systems (with R. Kotecky, D. Ueltschi). Commun. Math. Phys. 181, 409-446 (1996).