Phase separation in random cluster models I: uniform upper bounds on local deviation
This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the Fortuin-Kasteleyn planar random cluster model on the presence of an open dual circuit Gamma_0 encircling the origin and enclosing an area of at leas...
| المؤلف الرئيسي: | |
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| التنسيق: | Journal article |
| اللغة: | English |
| منشور في: |
2010
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| _version_ | 1826284153651331072 |
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| author | Hammond, A |
| author_facet | Hammond, A |
| author_sort | Hammond, A |
| collection | OXFORD |
| description | This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the Fortuin-Kasteleyn planar random cluster model on the presence of an open dual circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. (By the Fortuin-Kasteleyn representation, the model is a close relative of the droplet formed by conditioning the Potts model on an excess of spins of a given type.) We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Gamma_0), this being the maximum distance from a point in the circuit Gamma_0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma_0), namely, the length of the longest line segment of which the boundary of the convex hull is formed. The principal conclusion of the series of papers is the following uniform control on local deviation: that there are positive constants c and C such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) lies in the interval [c,C] tends to 1 in the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). In this way, we confirm the anticipated n^{1/3} scaling of maximum local roughness, and provide a sharp logarithmic power-law correction. This local deviation behaviour occurs by means of locally Gaussian effects constrained globally by curvature, and we believe that it arises in a range of radially defined stochastic interface models, including several in the Kardar-Parisi-Zhang universality class. This paper is devoted to proving the upper bounds in these assertions, and includes a heuristic overview of the surgical technique used in the three papers. |
| first_indexed | 2024-03-07T01:09:36Z |
| format | Journal article |
| id | oxford-uuid:8c8bea4c-922c-44d6-8887-6e5ea3fa35c9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:09:36Z |
| publishDate | 2010 |
| record_format | dspace |
| spelling | oxford-uuid:8c8bea4c-922c-44d6-8887-6e5ea3fa35c92022-03-26T22:45:11ZPhase separation in random cluster models I: uniform upper bounds on local deviationJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:8c8bea4c-922c-44d6-8887-6e5ea3fa35c9EnglishSymplectic Elements at Oxford2010Hammond, AThis is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the Fortuin-Kasteleyn planar random cluster model on the presence of an open dual circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. (By the Fortuin-Kasteleyn representation, the model is a close relative of the droplet formed by conditioning the Potts model on an excess of spins of a given type.) We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Gamma_0), this being the maximum distance from a point in the circuit Gamma_0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma_0), namely, the length of the longest line segment of which the boundary of the convex hull is formed. The principal conclusion of the series of papers is the following uniform control on local deviation: that there are positive constants c and C such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) lies in the interval [c,C] tends to 1 in the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). In this way, we confirm the anticipated n^{1/3} scaling of maximum local roughness, and provide a sharp logarithmic power-law correction. This local deviation behaviour occurs by means of locally Gaussian effects constrained globally by curvature, and we believe that it arises in a range of radially defined stochastic interface models, including several in the Kardar-Parisi-Zhang universality class. This paper is devoted to proving the upper bounds in these assertions, and includes a heuristic overview of the surgical technique used in the three papers. |
| spellingShingle | Hammond, A Phase separation in random cluster models I: uniform upper bounds on local deviation |
| title | Phase separation in random cluster models I: uniform upper bounds on
local deviation |
| title_full | Phase separation in random cluster models I: uniform upper bounds on
local deviation |
| title_fullStr | Phase separation in random cluster models I: uniform upper bounds on
local deviation |
| title_full_unstemmed | Phase separation in random cluster models I: uniform upper bounds on
local deviation |
| title_short | Phase separation in random cluster models I: uniform upper bounds on
local deviation |
| title_sort | phase separation in random cluster models i uniform upper bounds on local deviation |
| work_keys_str_mv | AT hammonda phaseseparationinrandomclustermodelsiuniformupperboundsonlocaldeviation |