×

zbMATH — the first resource for mathematics

On the spatial Markov property of soups of unoriented and oriented loops. (English) Zbl 1370.60192
Donati-Martin, Catherine (ed.) et al., Séminaire de probabilités XLVIII. Cham: Springer (ISBN 978-3-319-44464-2/pbk; 978-3-319-44465-9/ebook). Lecture Notes in Mathematics 2168. Séminaire de Probabilités, 481-503 (2016).
Let \(\Gamma\) be a directed graph with \(d(x)<\infty\) edges going out of each vertex \(x\in \Gamma\). A rooted \(n\)- steps loop \(l\) from a vertex \(l_0\) is \(l=(l_0,e_1,l_1,\dots,l_{n-1}, e_{n-1},l_n),\) where \(l_n=l_0\) and \(e_i\) is an edge from \(l_{i-1}\) to \(l_i\). Then, \[ p_l= \frac{1}{\prod_{i=0}^{n-1} d(l_i)} \] is the probability that the \(n\)-step trajectory of the random walk on \(\Gamma\) starting from \(l_0\) is exactly the loop \(l\). Next, on the set of all loops \(l\) of length \(n\) the measure \(\rho\) is defined by \(\rho_l=\frac{p_l}{n}.\) In a similar manner, a measure on the set of loops of an undirected graph is defined. The paper focuses on the description and the discussion of the spatial Markov property of the loops, accompanied by an analytical review of the relevant literature.
For the entire collection see [Zbl 1359.60007].

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
60G60 Random fields
60J27 Continuous-time Markov processes on discrete state spaces
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
05C81 Random walks on graphs
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] D. Brydges, J. Fröhlich, T. Spencer, The random walk representation of classical spin systems and correlation inequalities. Commun. Math. Phys. 83, 123–150 (1982)
[2] F. Camia, M. Lis, Non-backtracking loop soups and statistical mechanics on spin networks (2015). arXiv:1507.05065 · Zbl 1366.82020
[3] J. Dubédat, SLE and the free field: partition functions and couplings. J. Am. Math. Soc. 22, 995–1054 (2009) · Zbl 1204.60079
[4] E.B. Dynkin, Markov processes as a tool in field theory. J. Funct. Anal. 50, 167–187 (1983) · Zbl 0522.60078
[5] E.B. Dynkin, Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55, 344–376 (1984) · Zbl 0533.60061
[6] G.F. Lawler, Loop-erased random walk, in Perplexing Problems in Probability. Festschrift in Honor of Harry Kesten, Progress in Probability, vol. 44 (Birkhäuser, Boston, 1999), pp. 197–217
[7] G.F. Lawler, V. Limic, Random Walks: a Modern Introduction (Cambridge University Press, Cambridge, 2010) · Zbl 1210.60002
[8] G.F. Lawler, W. Werner, The Brownian loop soup. Probab. Theory Relat. Fields 128, 565–588 (2004) · Zbl 04576228
[9] Y. Le Jan, Markov Paths, Loops and Fields. Lecture Notes in Mathematics, vol. 2026 (Springer, Berlin, 2011) · Zbl 1231.60002
[10] T. Lupu, From loop clusters and random interlacement to the free field. Ann. Probab. 44 (3), 2117–2146 (2016) · Zbl 1348.60141
[11] T. Lupu, W. Werner, A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field. Electron. Commun. Probab. 21 (2016) · Zbl 1338.60236
[12] M.B. Marcus, J. Rosen, Markov Processes, Gaussian Processes, and Local Times (Cambridge University Press, Cambridge, 2006) · Zbl 1129.60002
[13] E. Nelson, The free Markoff field. J. Funct. Anal. 12, 211–227 (1973) · Zbl 0273.60079
[14] W. Qian, W. Werner, Decomposition of two-dimensional loop-soup clusters (2015). arXiv:1509.01180
[15] O. Schramm, S. Sheffield, A contour line of the continuous Gaussian free field. Probab. Theory Relat. Fields 157, 47–80 (2013) · Zbl 1331.60090
[16] S. Sheffield, W. Werner, Conformal loop ensembles: the Markovian characterization and the loop-soup construction. Ann. Math. 176, 1827–1917 (2012) · Zbl 1271.60090
[17] K. Symanzik, Euclidean quantum field theory, in Local Quantum Theory, ed. by R. Jost (Academic, New York, 1969)
[18] W. Werner, SLEs as boundaries of clusters of Brownian loops. C.R. Math. Acad. Sci. Paris 337, 481–486 (2003) · Zbl 1029.60085
[19] W. Werner, Topics on the Gaussian Free Field. Lecture Notes (Springer, Berlin, 2014)
[20] D.B. Wilson, Generating random spanning trees more quickly than the cover time, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing (Association for Computing Machinery, New York, 1996), pp. 296–303 · Zbl 0946.60070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.