Alan Hammond
Associate Professor
Departments of Mathematics and Statistics
University of California at Berkeley
899 Evans Hall
Berkeley, CA
94720-3840
USA
Email: :
Leave to remain?
Vladimir:
Well? Shall we go?
Estragon:
Yes, let's go.
They do not move.
The story of the skylark,
Pacsirta, in a few words and pictures.
Research
My research concerns probability theory, statistical mechanics and partial differential equations:
a research overview.
These Minerva lectures ,
given at Columbia University in February and March 2019, offer an overview of the use of probabilistic tools in analysing scaling limits of local models of random growth in the Kardar-Parisi-Zhang universality class.
During the autumn of 2012, I gave a graduate course at the University of Geneva concerning the kinetic limit derivation of Smoluchowski's coagulation-diffusion PDE.
This article developed from notes from the class:
Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDE.
Probab. Surv., 14 , 205--288 (2017).
Publications
Critical exponents in percolation via lattice animals .
Electron. Comm. Probab., 10 , no.4, 45--59 (2005).
Fluctuation of planar Brownian loop capturing large area .
With Yuval Peres.
Trans. Amer. Math. Soc., 360 , no. 12, 6197--6230 (2008).
The kinetic limit of a system of coagulating Brownian particles .
With Fraydoun Rezakhanlou.
Arch. Rational Mech. Anal., 185 , 1--67 (2007).
Kinetic limit for a system of coagulating planar Brownian particles .
With Fraydoun Rezakhanlou.
J. Stat. Phys., 124 , 997--1040 (2006).
Moment bounds for the Smoluchowski equation and their consequences .
With Fraydoun Rezakhanlou.
Comm. Math. Phys., 276 , no. 3, 645--670 (2007).
Greedy lattice animals: geometry and
criticality .
Ann. Probab., 34 , no.2, 593--637, (2006).
Coagulation, diffusion and the continuous
Smoluchowski equation .
With Mohammad Reza Yaghouti and Fraydoun Rezakhanlou.
Stochastic Process. Appl., 119 , no. 9, 3042--3080, (2009).
Monotone loop models and rational resonance .
With Richard Kenyon.
Probab. Theory and Related Fields,
150 , no. 3-4 ,613--633, (2011)
Biased random walks on Galton-Watson trees with
leaves .
With Gerard Ben Arous, Alexander Fribergh and Nina Gantert.
Ann. Probab., 40 , no. 1, 280--338, (2012).
Power-law Polya's urn and fractional Brownian motion .
With Scott Sheffield.
Probab. Theory and Related Fields, 157 , no. 3, 691--719, (2013).
Phase separation in random cluster models I:
uniform upper bounds on local deviation .
Comm. Math Phys., 310 , no. 2, 455--509, (2012)
Phase separation in random cluster models II:
the droplet at equilibrium, and local deviation lower bounds .
Ann. Probab., 40 , no. 3, 921--978, (2012)
Phase separation in random cluster models III:
circuit regularity .
J. Stat. Phys., 142 , no. 2, 229--276, (2011)
Randomly biased walks on subcritical trees .
With Gerard Ben Arous.
Comm. Pure Appl. Math., 65 , no. 11, 1481--1527, (2012)
Stable limit laws for randomly biased walks on supercritical trees .
Ann. Probab., 41 , no. 3A, 1694--1766, (2013)
Exit time tails from pairwise decorrelation in hidden Markov chains, with applications to dynamical percolation .
With Elchanan Mossel and Gabor Pete.
Electron. J. Probab., 17 , article 68, 1--16, (2012)
Phase transition for the speed of the biased random walk on the supercritical percolation cluster .
With Alex Fribergh.
Comm. Pure Appl. Math., 67 , no. 2, 173--245, (2014)
Infinite cycles in the random stirring model on trees .
Bulletin of the Institute of Mathematics, Academia Sinica, Special Issue in honour of S.R.S. Varadhan's 70th birthday, 8 , no. 1, 85--104, (2013)
Self-avoiding walk is sub-ballistic .
With Hugo Duminil-Copin.
Comm. Math. Phys., 324 , no. 2, 401--423, (2013)
Brownian Gibbs property for Airy line ensembles .
With Ivan Corwin.
Invent. Math., 195 , 441--508, (2014)
Sharp phase transition in the random stirring model on trees .
Probab. Theory and Related Fields, 161 , no. 3-4, 429--448, (2015)
Local time on the exceptional set of dynamical percolation, and the Incipient Infinite Cluster .
With Gabor Pete and Oded Schramm.
Ann. Probab., 43 , no. 6, 2949--3005, (2015)
On the probability that self-avoiding walk ends at a given point .
With Hugo Duminil-Copin, Alexander Glazman and Ioan Manolescu.
Ann. Probab., 44 , no. 2, 955--983, (2016)
KPZ Line Ensemble .
With Ivan Corwin.
Probab. Theory and Related Fields, 166 , no. 1-2, 67--185, (2016)
The competition of roughness and curvature in area-constrained polymer models .
With Riddhipratim Basu and Shirshendu Ganguly.
Comm. Math. Phys.,
364 , no. 3, 1121--1161, (2018)
An upper bound on the number of self-avoiding polygons via joining .
Ann. Probab., 46 , no. 1, 175--206, (2018)
On self-avoiding polygons and walks: the snake method via pattern fluctuation .
Trans. Amer. Math. Soc., to appear.
On self-avoiding polygons and walks: the snake method via polygon joining .
Electron. J. Probab., to appear.
The next article gathers together the content of the preceding three; it includes some exposition and emphasises some common themes.
On self-avoiding polygons and walks: counting, joining and closing .
Online survey.
Critical point for infinite cycles in a random loop model on trees . With Milind Hegde.
Ann. Appl. Probab., to appear.
Self-attracting self-avoiding walk . With Tyler Helmuth.
Probab. Theory and Related Fields, to appear.
Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation .
Mem. Amer. Math. Soc., to appear.
Modulus of continuity of polymer weight profiles in Brownian last passage percolation .
Ann. Probab., to appear.
Exponents governing the rarity of disjoint polymers in Brownian last passage percolation .
Proc. Lond. Math. Soc., to appear.
A patchwork quilt sewn from Brownian fabric:
regularity of polymer weight profiles in Brownian last passage percolation
.
Forum Math. Pi, 7 (2019), e2. 69 pp.
Bounding the number of self-avoiding walks: Hammersley-Welsh with polygon insertion . With Hugo Duminil-Copin, Shirshendu Ganguly and Ioan Manolescu.
Ann. Probab., to appear.
Preprints