I am broadly interested in questions that have a probabilistic or analytic flavor, and take much inspiration from physics as an inexhaustible source of interesting questions and mathematical problems. Such problems often concern models which have the stucture that many simple constituents (particles, spins, graphs, and so on) interact with other spatially close consituents in a simple fashion. The main questions then concern the global collective behavior of all particles.
In particular, one of the main themes in my research has been the understanding of the interaction of many length scales and the emergence of universal behaviour in different contexts, and related to this, the mathematical implementation of the renormalisation group method. Numerous related problems and apparently rather disparate models can be formulated in terms of statistical field theory, interpreted in a broad sense, and the use and exploration of this idea is an important influence in my research.
Spin systems like the Ising or the O(n) models are fundamental models for phase transitions, providing the perhaps simplest representatives in their conjecturally broad universality classes. Much progress has been made in the understanding of spin systems over the past century, by making use of a vast range of mathematical tools, yet many fundamental aspects have remained elusive. My interest in such systems is in particular in those with continuous symmetry, which are the least understood. For example, a robust understanding of their phase transitions is still mostly lacking, and the subtle problem of `mass generation’ in the marginal dimension remains one of the major problems of mathematical physics. My past research on spin systems has focused in particular on the Ginzburg–Landau O(n) model (the φ4 model) and the renormalisation group approach. An overview is given in my book with Brydges and Slade.
Stochastic dynamics of spin and particle systems, i.e., the evolution of such systems by local dynamics with noise (such as Glauber and Kawasaki dynamics), provides a fundamental test case for the understanding of statistical physics out of equilibrium. Compared with the equilibrium situation, the mathematical understanding of non-equilibrium statistical physics remains far from developed. Not only are many of the tools that have proved to be powerful for equilibrium systems lost out of equilibrium, but in fact related to this, there is a host of new phenomena associated to dynamics without static counterparts. My research in this field has focused on developing renormalisation group ideas for the study of spectral gaps and log-Sobolev constants which provide key control over the dynamics.
Supersymmetry originated as the idea of a symmetry between Bosons and Fermions in particle physics, but surprisingly, it has also become a powerful idea from the point of view of statistical physics and condensed matter physics. There it relates to various fundamental models with no apparent connection to particle physics, often involving disorder. Such systems include, among others, random matrices and operators, stochastic differential equations, self-avoiding walks, reinforced random walks, random spanning forests, and branched polymers. My research has explored these connections, for example, in the study of self-avoiding walks, reinforced walks, or random spanning forests, and exploited them to prove otherwise inaccessible results about the former probabilistic models.
Coulomb systems take a central position in statistical physics. As particle systems, they are not only the hardest models to study analytically, due to their singular long-range forces, but it is exactly this singular behaviour that makes these models most interesting. It comes with new phenomena such as screening of particles or rigidity are a consequence—some of which are understood though many have remained elusive. At the same time, such Coulomb systems are related in intricate ways to the statistical physics of apparently unrelated models, including the XY model (for interaction of vortices), in Conformal Field Theory (in the Coulomb gas formalism), in the problem of crystallisation (in the interaction of crystal dislocations), or in random matrix theory (in the interaction of eigenvalues). My research has touched upon various aspects of Coulomb systems, including the fluctuations of the one-component plasma, the study of various aspects of the sine-Gordon field theory, or the crystallisation problem. Furthermore, Coulomb systems continue to provide a source of inspiration for many future problems.
Random matrices were motivated by Wigner as simple models for disordered quantum systems. Now they play an important role not only in statistical physics, but also number theory, the theory of integrable systems, computer science, graph theory, statistics, and other areas of mathematics. I have been particularly interested in the example of the random regular graph.
- Log-Sobolev inequality for near critical Ising models R. Bauerschmidt, B. Dagallier Preprint
- Log-Sobolev inequality for the φ42 and φ43 measures R. Bauerschmidt, B. Dagallier Preprint
- The Discrete Gaussian model, II. Infinite-volume scaling limit at high temperature R. Bauerschmidt, J. Park, P.-F. Rodriguez Preprint
- The Discrete Gaussian model, I. Renormalisation group flow at high temperature R. Bauerschmidt, J. Park, P.-F. Rodriguez Preprint
- Percolation transition for random forests in d ≥ 3 R. Bauerschmidt, N. Crawford, T. Helmuth Preprint
- The Coleman correspondence at the free fermion point R. Bauerschmidt, C. Webb J. Eur. Math. Soc., to appear
- Maximum and coupling of the sine-Gordon field R. Bauerschmidt, M. Hofstetter Ann. Probab., 50, 455–508, (2022)
- Random Spanning Forests and Hyperbolic Symmetry R. Bauerschmidt, N. Crawford, T. Helmuth, A. Swan Commun. Math. Phys., 381, 1223–1261, (2021)
- Mean-field tricritical polymers R. Bauerschmidt, G. Slade Probab. Math. Phys., 1, 167–204, (2020)
- Edge rigidity and universality of random regular graphs of intermediate degree R. Bauerschmidt, J. Huang, A. Knowles, H.-T. Yau Geom. Funct. Anal., 30, 693–769, (2020)
- Log-Sobolev inequality for the continuum sine-Gordon model R. Bauerschmidt, T. Bodineau Comm. Pure Appl. Math., 74, 2064–2113, (2021)
- Three-dimensional tricritical spins and polymers R. Bauerschmidt, M. Lohmann, G. Slade J. Math. Phys., 61, 033302, 30, (2020)
- The geometry of random walk isomorphism theorems R. Bauerschmidt, T. Helmuth, A. Swan Ann. Inst. Henri Poincaré Probab. Stat., 57, 408–454, (2021)
- Spectral Gap Critical Exponent for Glauber Dynamics of Hierarchical Spin Models R. Bauerschmidt, T. Bodineau Commun. Math. Phys., 373, 1167–1206, (2020)
- The two-dimensional Coulomb plasma: quasi-free approximation and central limit theorem R. Bauerschmidt, P. Bourgade, M. Nikula, H.-T. Yau Adv. Theor. Math. Phys., 23, 841–1002, (2019)
- Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process R. Bauerschmidt, T. Helmuth, A. Swan Ann. Probab., 47, 3375–3396, (2019)
- Dislocation Lines in Three-Dimensional Solids at Low Temperature R. Bauerschmidt, D. Conache, M. Heydenreich, F. Merkl, S.W.W. Rolles Ann. Henri Poincaré, 20, 3019–3057, (2019)
- Local Kesten-McKay law for random regular graphs R. Bauerschmidt, J. Huang, H.-T. Yau Commun. Math. Phys., 369, 523–636, (2019)
- A very simple proof of the LSI for high temperature spin systems R. Bauerschmidt, T. Bodineau J. Funct. Anal., 276, 2582–2588, (2019)
- Bulk eigenvalue statistics for random regular graphs R. Bauerschmidt, J. Huang, A. Knowles, H.-T. Yau Ann. Probab., 45, 3626–3663, (2017)
- Local density for two-dimensional one-component plasma R. Bauerschmidt, P. Bourgade, M. Nikula, H.-T. Yau Commun. Math. Phys., 356, 189–230, (2017)
- Local semicircle law for random regular graphs R. Bauerschmidt, A. Knowles, H.-T. Yau Comm. Pure Appl. Math., 70, 1898–1960, (2017)
- Four-dimensional weakly self-avoiding walk with contact self-attraction R. Bauerschmidt, G. Slade, B.C. Wallace J. Stat. Phys., 167, 317–350, (2017) (Errata)
- Finite-order correlation length for four-dimensional weakly self-avoiding walk and |φ|4 spins R. Bauerschmidt, G. Slade, A. Tomberg, B.C. Wallace Ann. Henri Poincaré, 18, 375–402, (2017) (Errata)
- Critical two-point function of the 4-dimensional weakly self-avoiding walk R. Bauerschmidt, D.C. Brydges, G. Slade Commun. Math. Phys., 338, 169–193, (2015)
- Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis R. Bauerschmidt, D.C. Brydges, G. Slade Commun. Math. Phys., 337, 817–877, (2015) (Errata)
- A renormalisation group method. III. Perturbative analysis R. Bauerschmidt, D.C. Brydges, G. Slade J. Stat. Phys., 159, 492–529, (2015) (Errata) (Code)
- Structural stability of a dynamical system near a non-hyperbolic fixed point R. Bauerschmidt, D.C. Brydges, G. Slade Ann. Henri Poincaré, 16, 1033–1065, (2015) (Errata)
- Scaling limits and critical behaviour of the 4-dimensional n-component |φ|4 spin model R. Bauerschmidt, D.C. Brydges, G. Slade J. Stat. Phys., 157, 692–742, (2014) (Errata)
- Fluctuations in a kinetic transport model for quantum friction R. Bauerschmidt, W. Roeck, J. Fröhlich J. Phys. A, 47, 275003, 15, (2014)
- A simple method for finite range decomposition of quadratic forms and Gaussian fields R. Bauerschmidt Probab. Theory Related Fields, 157, 817–845, (2013)
- Introduction to a renormalisation group method R. Bauerschmidt, D.C. Brydges, G. Slade Lecture Notes in Mathematics, 2242, xii+281, (2019)
Proceedings and lecture notes
- Spin systems with hyperbolic symmetry: a survey R. Bauerschmidt, T. Helmuth Proceedings ICM 2022
- Renormalisation group analysis of 4D spin models and self-avoiding walk R. Bauerschmidt, D.C. Brydges, G. Slade Proceedings ICMP 2015
- Lectures on self-avoiding walks R. Bauerschmidt, H. Duminil-Copin, J. Goodman, G. Slade Clay Math. Proc., 15, 395–467, (2012)