Book in progress
R.J. Adler, J.E. Taylor, (2025/6) Random Fields and Geometry: Towards Applications
This book is almost finished, and was written with statisticians and other researchers interested in applications in mind. It contains a brief review of the theory of Random Fields and Geometry , with considerably different emphasis and additions to, and versions of, that theory that are of practical importance. More significantly, it goes into detail as to how to apply the theory in practice, along with a slew of examples.
The version posted here (July 22, 2025) here , is still missing one rather important chapter, but is otherwise complete. This does not mean that it has been fully debugged, so you should expect some errors and oversights.
We shall be very grateful for all comments, questions, suggestions and lists of typos that will help improve the final version.
21-st century books
- R.J. Adler and J.E. Taylor, (2011), Topological complexity of smooth random functions. These are notes of Saint Flour Lectures, 2009, and and an early version is avaialble here.
- R.J. Adler and J.E. Taylor, (2007), Random Fields and Geometry. Go to here (for Europe) or here (for USA) for details. It is written in three parts, and you can get some more details from the following excerpts: Preface, Contents, Part I: Gaussian processes, Part II: Geomerty, PART III: Geometry of random fields, Index. For preprints of specific chapters, please write directly to me. There is also a correction list .
20-th century books
- R.J. Adler, (1981), The Geometry of Random Fields , Wiley, London. xi+275, reprinted by SIAM, 2010 in their Classics in Applied Mathematics series, and this despite the fact that, unlike most of their authors, I am still alive.
- R.J. Adler, (1990), An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, IMS Lecture Notes-Monograph Series, Vol 12, vii + 160. Corrections to Chapter 2 can be found here .
- R.J. Adler, P. Muller, and B. Rozovskii, (eds), Stochastic Modelling in Physical Oceanography, Birkhaüser, Boston.
- R.J. Adler, R. Feldman, M. Taqqu, (eds) (1998), A Users Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tailed Distributions and Processes, Birkhaüser, Boston.
(With thanks to a grandson and Chatgpt)
Some recent preprints
Actually, “recent” is not necessarily all that recent, but refers to when I started making hyperlinked lists. For a full (modulo the papers that are so new even Google has not caught them yet) list, you can go to my Google Scholar page (but you already knew that anyway).
- R.Feng, D. Yao, and R.J. Adler, Critical radii and suprema of random waves over Riemannian manifolds
- E. Onaran, O. Bobrowski and R.J. Adler, Functional central limit theorems for local functionals of dynamic point processes,
- E. Onaran, O. Bobrowski and R.J. Adler, Functional central limit theorems for local statistics of spatial birth-death processes in the thermodynamic regime
- K. Adhikari, R.J. Adler, O. Bobrowski and R. Rosenthal, On the spectrum of dense random graphs
- P. Pranav, R.J. Adler, T. Buchert, H. Edelsbrunner, B.J.T. Jones, A. Schwartzman, H. Wagner, and R. van de Weygaert, Unexpected Topology of the Temperature Fluctuations in the Cosmic Microwave Background
- P. Pranav, R. van de Weygaert, G. Vegter, B.J.T. Jones, R.J. Adler, J. Feldbrugge, C. Park, T. Buchert and M. Kerber, Topology and Geometry of Gaussian random fields I: on Betti Numbers, Euler characteristic and Minkowski functionals
- R.J. Adler and S. Agami, Modelling persistence diagrams with planar point processes, and revealing topology with bagplots.
- R. Feng, X. Xu and R.J. Adler Critical radius and supremum of random spherical harmonics (II)
- S. Agami and R.J. Adler Modelling of persistent homology
- R.J. Adler, S. Agami and P. Pranav, Modeling and replicating statistical topology, and evidence for CMB non-homogeneity.
- R.J. Adler, S. Agami and P. Pranav, Supplementary Information Appendix for Modeling and replicating statistical topology, and evidence for CMB non-homogeneity.
- R.J. Adler, K. Bartz, S. Kou, and A. Monod, Estimating thresholding levels for random fields via Euler characteristics.
- R. Feng and R.J. Adler, Critical radius and supremum of random spherical harmonics
- S.R. Krishnan, J.E. Taylor and R.J. Adler The intrinsic geometry of some random manifolds
- G. Naitzat and R.J. Adler, A central limit theorem for the Euler integral of a Gaussian random field
- T. Owada and R.J. Adler, Limit theorems for point processes under geometric constraints (and topological crackle)
- R.J. Adler, S.R. Krishnan, J.E. Taylor and S. Weinberger, Convergence of the reach for a sequence of Gaussian-embedded manifolds
- G. Thoppe, D.\ Yogeshwaran and R.J. Adler, On the evolution of topology in dynamic Erdos-Renyi graphs
- R.J. Adler and G. Samorodnitsky, Climbing down Gaussian peaks
- D. Yogeshwaran, E. Subag and R.J. Adler, Random geometric complexes in the thermodynamic regime
- R.J. Adler, O. Bobrowski and S. Weinberger, Crackle: The persistent homology of noise
- D. Yogeshwaran and R.J. Adler, On the topology of random processes built over stationary random processes
- R.J. Adler, E. Moldavskaya and G. Samorodnitsky, On the existence of paths between points in high level excursion sets of Gaussian random fields.
- R.J. Adler, E. Subag and J.E. Taylor, Rotation and scale space random fields and the Gaussian kinematic formula
- O. Bobrowski and R.J. Adler, Distance functions, critical points, and topology for some random complexes.
- A. Schwartzman, Y. Gavrilov, P. Jadinsky and R.J. Adler, Multiple testing of local maxima for detection of peaks in 1D.
- R.J. Adler, J. Blanchet and J. Liu, Efficient Monte Carlo for high excursions of Gaussian random fields.
- R.J. Adler, O. Bobrowski, M.S. Borman, E. Subag and S. Weinberger, Persistent homology for random fields and complexes
- R.J. Adler, G. Samorodnitsky and J.E. Taylor, High level excursion set geometry for non-Gaussian infinitely divisible random fields
- R.J. Adler, J. Ewing and P. Taylor, Citation Statistics: A report from the International Mathematical Union in cooperation with the International Council of Industrial and Applied Mathematics and the Institute of Mathematical Statistics
- R.J. Adler, J. Blanchet and J. Liu, Efficient simulation for tail probabilities of Gaussian random fields
- R.J. Adler, Some new random field tools for spatial analysis
- R.J. Adler, G. Samorodnitsky and J.E. Taylor, Excursion sets of stable random fields
- J.E. Taylor and R.J. Adler, Gaussian processes, kinematic formulae and Poincare’s limit
- G. Bonnet and R.J. Adler, The Burgers superprocess
- S. Vadlamani and R.J. Adler, Global geometry under isotropic Brownian flows
- A. Zeevi, R. Meir and R.J. Adler, Non-linear models for time series using mixtures of autoregressive models
Movie and Computer Package
- Superprocesses: The Movie. (should be uploaded to youtube) A 15-minute video of superprocess visualisations. The movie was produced at the Technion Visualization Centre . Drop me a line at radler@technion.ac.il and I will mail you a CD with all the mpeg files, or download some now, for simulations of simple superprocesses in one, two, or three dimensions. (In each one of these simulations you will first see a simulation of many particles following a simple random walk, with total mass spreading out according to the discrete heat equation with small random perturbation, followed by branching random walks meant to simulate superprocesses.)
- SUPER, an interactive C/GL graphics package for the visualisation of superprocesses in dimensions one through five.Documentation, including information on how to obtain, run, and, if necessary, compile the programs, is here.