Random Walks and Physical Fields

Yves Le Jan is a French mathematician working in Probability theory and Stochastic processes. In 2006 he was invited speaker at the International Congress of Mathematicians in Madrid. In 2008 he became Senior Member of the Institut Universitaire de France. In 2011 he was Doob Lecturer at the 8th World Congress in Probability and Statistics in Istanbul. In 1995 he was awarded the Poncelet Prize and in 2011 the Sophie Germain Prize of the French Academy of Sciences. He is Professor emeritus at the Orsay Institute of Mathematics of Université Paris-Saclay and since 2021 visiting Professor at NYUAD.​

“I enjoyed reading this very well-written and interesting book. It introduces objects such as Markov loops, spanning forests, random homologies and covers, which are related to random walks on graphs. Relations between these objects and random fields used in mathematical physics are explored.” (John Masson Noble, Mathematical Reviews, May, 2025)

• 1 Markov Chains and Potential Theory on Graphs.- 2 Loop Measures.- 3 Decompositions, Traces and Excursions.- 4 Occupation Fields.- 5 Primitive Loops, Loop Clusters, and Loop Percolation.- 6 The Gaussian Free Field.- 7 Networks, Ising Model, Flows, and Configurations.- 8 Loop Erasure, Spanning Trees and Combinatorial Maps.- 9 Fock Spaces, Fermi Fields, and Applications.- 10 Groups and Covers.- 11 Holonomies and Gauge Fields.- 12 Reflection Positivity and Physical Space.