Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 15 (2018) open journal systems 

Topics in loops measures and the loop-erased walk

Gregory F. Lawler, University of Chicago
Full Text: PDF

Lawler, Gregory F., Topics in loops measures and the loop-erased walk, Probability Surveys, 15, (2018), 28-101 (electronic). DOI: 10.1214/17-PS293.


[1]    M. Aizenman, H. Duminil-Copin, and V. Sidoravicius (2014). Random currents and continuity of Ising model’s spontaneous magnetization, Commun. Math. Phys. 334, 719–742. MR3306602

[2]    P. Berg, J. McGregor (1966). Elementary Partial Differential Equations, Holden-Day. MR0201770

[3]    I. Benjamini, H, Kesten, Y. Peres, O. Schramm (2004). Geometry of the uniform spanning forest: Transitions in dimensions 4,8,12,.., Annals. Math 160, 465–491. MR2123930

[4]    C. Beneš, G. Lawler, F. Viklund. Scaling limit of the loop-erased random walk Green’s function, to appear in Probab. and Related Fields. MR3547740

[5]    S. Fomin (2001). Loop-erased walks and total positivity, Trans. Am. Math. Soc., 353(9):3563–3583. MR1837248

[6]    A, Karilla, K, Kytölä, E. Peltola (2017). Boundary correlations in planar LERW and UST, preprint.

[7]    R. Kenyon (2000). The asymptotic determinant of the discrete Laplacian. Acta Math., 239–286. MR1819995

[8]    R. Kenyon, D. Wilson (2011). Double dimer pairings and skew Young diagrams, Electron. J. of Combin. 18, paper #P130. MR2811099

[9]    M. Kozdron, G. Lawler (2005). Estimates of random walk exit probabilities and application to loop-erased walk, Electron. J. of Probab. 10 paper no. 44. MR2191635

[10]    G. Lawler (1980). A self-avoiding random walk, Duke Mathematical Journal 47, 655–694. MR0587173

[11]    G. Lawler (1983). A connective constant for loop-erased self-avoiding random walk, J. Appl. Prob. 20, 264–276. MR0698530

[12]    G. Lawler (1999). Loop-erased random walk, in Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten, M. Bramson and R. Durrett, ed., Birkhäuser-Boston, 197–217. MR1703133

[13]    G. Lawler (2014). The probability that loop-erased random walk uses a given edge, Electron. Comm. Probab. 19, article no. 51. MR3246970

[14]    G. Lawler, O. Schramm. and W. Werner (2003). Conformal restriction: the chordal case, J. Amer. Math. Soc. 16, 917–955. MR1992830

[15]    G. Lawler, J. A. Trujillo Ferreras (2007). Random walk loop soup, Trans. Amer. Math. Soc. 359, 767–787. MR2255196

[16]    G. Lawler, W. Werner (2004), The Brownian loop soup, Probab. Theory Related Fields 128, 565–588. MR2045953

[17]    G. Lawler and V. Limic (2010). Random Walk: A Modern Introduction, Cambridge U. Press. MR2677157

[18]    G. Lawler and J. Perlman (2015). Loop measures and the Gaussian free field, in Random Walks, Random Fields, and Disordered Systems, Lecture Notes in Mathematics 2144, M. Biskup, J. Černý, R. Kotecký, ed., Springer-Verlag, 211–235. MR3382175

[19]    G. Lawler, X. Sun, W. Wu (2016). Loop-erased random walk, uniform spanning forests, and bi-Laplacian Gaussian field, preprint. MR1703133

[20]    Y. Le Jan (2011). Markov Paths, Loops, and Fields, Lecture Notes in Mathematics 2026, Springer-Verlag. MR2815763

[21]    T. Lupu (2016). From loop clusters and random interlacements to the free field, Annals of Probab. 44, 2117–2146. MR3502602

[22]    T. Lupu, W, Werner (2016). A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field, Electron. Commun in Probab. 21, paper no. 7. MR3485382

[23]    S. S. Manna, D. Dhar, S. Majumdar (1992). Spanning trees in two dimensions, Phys. Rev. A 46, R4471(R).

[24]    P. Marchal (2000). Loop-Erased random walks, spanning trees and hamiltonian cycles 5, paper no, 4, 39–50. MR1736723

[25]    R. Pemantle (1991). Choosing a spanning tree for the integer lattice uniformly, Ann. Probab. 19, 1559–1574. MR1127715

[26]    M. Picadello and W. Weiss, ed., (1999). Random Walks and Discrete Potential Theory, conference proceedings from Cortona 1997, Cambridge U. Press. MR1802423

[27]    L. Rosen (1996). Positive powers of positive positive definite matrices, Can. J. Math. 48, 196–209. MR1382482

[28]    O. Schramm (2000). Scaling limits of loop-erased random walk and uniform spanning trees, Israel J. Math. 118, 221–288. MR1776084

[29]    D. Wilson (1996). Generating random spanning trees more quickly than the cover time, Proc. STOC96, 296–303.

Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787