This book offers an in-depth review of kinetically constrained models (KCMs), a topic that lies at the crossroads of probability and statistical mechanics. KCMs have captivated physicists ever since their introduction in the 1980s. Their remarkable glassy behavior makes them an essential toy model for exploring the liquidglass transition, a longstanding puzzle in condensed matter physics. Over the past 20 years, KCMs have also gained significant attention in mathematics. Despite belonging to the well-established domain of interacting particle systems with stochastic dynamics, the presence of dynamical constraints gives rise to novel phenomena. These include anomalously long mixing times, aging effects, singularities in the dynamical large deviation function, dynamical heterogeneities, and atypical ergodicity-breaking transitions corresponding to the emergence of a large variety of amorphous structures.
Authored by two leading experts in the field, this volume offers an extensive overview of rigorous results in the field. The self-contained exposition, with emphasis on high-level ideas and common techniques, is suitable for novices, as well as seasoned researchers, with backgrounds in mathematics or physics. The text covers crucial connections to bootstrap percolation cellular automata, along with sharp thresholds, universality, out-of-equilibrium dynamics, and more. The volume features challenging open questions and a detailed bibliography to direct future research. Whether as a reference or a study guide, it is a valuable resource for those interested in KCM.
Preface.- The models.- Setting and notation.- The Markov processes:
kinetically constrained spin models and kinetically constrained lattice
gases.- The most studied choices of constraints.- Some useful classification:
oriented/nonoriented models, cooperative/noncooperative models.-
Motivations from physics.- A crash course on liquid/glass and jamming
transitions.- The quest of the ideal glass transition: models on Bethe
lattices and the spiral model.- Kinetically Constrained Spin Models: the
basic results.- Ergodicity and connection with bootstrap percolation.-
Exponential convergence to equilibrium in L2.- The failure of classic
coercive inequalities (logarithmic and modified logarithmic Sobolev
constant).- Persistence and exchange times.- Scaling with density of the
spectral gap: the case of Friedrickson-Andersen 1f model.- Some open
problems.- Kinetically Constrained Spin Models on trees.- A martingale
technique to prove positivity of the spectral gap.- Power law scaling at
criticality.- An open problem.- The out of equilibrium regime.- An easy
perturbative result in one dimension.- Oriented models: East and models on
trees.- Non cooperative models.- Some open problems.- Dynamical phase
transition.- Activity and its large deviations.- The one dimensional case:
finite size effects and surface tension.- Open problems.- The East model.-
Combinatorics.- Spectral gap and mixing time.- Time scale separation.- Front
motion and cut-off.- Plateau behavior, aging and scaling limits.- The
generalized East process in higher dimensions.- An open problem: Aldous
Diaconis conjecture.- Kinetically Constrained Lattice Gases.- Ergodicity.-
Non cooperative models: spectral gap, log-Sobolev, tagged particle and
hydrodynamic limit.- Cooperative models: spectral gap and polynomial decay to
equilibrium.
Ivailo Hartarsky graduated from lycée Louis le Grand, then began research on bootstrap percolation while studying at the École Normale Supérieure in Paris. He went on to obtain his Ph.D. from Paris Dauphine University under the supervision of Cristina Toninelli in 2022 and conducted his postdoctoral research at TU Wien. As an associate researcher at Claude Bernard Lyon 1 University, the French National Centre for Scientific Research (CNRS) since 2024, Dr. Hartarskys work still focuses on bootstrap percolation, kinetically constrained models, and the interactions between them. More broadly, he is interested in cellular automata, interacting particle systems, and percolation. Some of his main contributions, such as sharp thresholds and universality, are covered in this book. He is also involved in extracurricular education of secondary school students in Bulgaria and international student mobility programs.
Cristina Toninelli received her masters and Ph.D. in theoretical physics from the University of Rome La Sapienza, then turned to mathematics, focusing on problems at the intersection of probability and statistical mechanics. In 2006, she joined the CNRS as a researcher, working at the Probability and Statistics Laboratory of Sorbonne and Paris Cité University. In 2018, she became a CNRS research director and moved to Ceremade at Paris Dauphine UniversityPSL. Since 2022, she has also been a professor at the École Normale Supérieure in Paris. Over her career, she has authored more than 50 publications in international journals, supervised numerous Ph.D. students, and led a European Research Council (ERC) grant team. She currently serves as the editor-in-chief of the Electronic Journal of Probability and was awarded the Marc Yor Prize by the Académie des Sciences for her contributions to kinetically constrained models.
