東華大學圖書館 |

The Geometry of Fronts and Interfaces Arising From Interacting Particle Systems /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	The Geometry of Fronts and Interfaces Arising From Interacting Particle Systems // Yujin Hong Kim.
作者:	Kim, Yujin Hong,
面頁冊數:	1 electronic resource (601 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
Contained By:	Dissertations Abstracts International86-12B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31931334
ISBN:	9798286424122

The Geometry of Fronts and Interfaces Arising From Interacting Particle Systems /
Kim, Yujin Hong,

The Geometry of Fronts and Interfaces Arising From Interacting Particle Systems /Yujin Hong Kim. - 1 electronic resource (601 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: B.

This work is primarily concerned with models of interfaces arising in large systems of interacting particles. Studying these objects will lead us to consider the Solid-on-Solid model, line ensembles of attracting and repelling random walks, branching Brownian motion, traveling-wave solutions to reaction-diffusion equations, the log-correlated random wave model (or random Fourier series), and Gaussian and non-Gaussian multiplicative chaos. This thesis consists of two parts.Part I examines the entropic repulsion phenomena of the (2 + 1)-dimensional Solid-on-Solid (SOS) model at low temperature above a hard wall, with zero boundary conditions. The SOS model is a canonical model for interfaces in equilibrium systems exhibiting phase coexistence. When considered in a box of side-length \uD835\uDC3F, Caputo et al. (2016) established that this random surface rises to a height of size \uD835\uDD25 ≍ log L (in contrast to its no-floor counterpart, which typically lies at O(1) height) via a series of slabs, or "levels," of height 1. The borders of these levels, called level lines, form concentric loops when viewed from above, and these loops admit a deterministic scaling limit when the box is shrunk to unit scale. The inner-most level lines are believed to exhibit a 1:2:3 scaling relation away from the corners of the box: that is, fluctuations of size L1/3 on intervals of length \uD835\uDC3F2/3 away from the limit shape.Our first result, joint with P. Caddeo and E. Lubetzky and presented in Chapter 2, is to establish the scaling relation in the sense of a lower bound: we show that, with high probability, the innermost level line has fluctuations of size at least order \uD835\uDC3F1/3 on intervals of length \uD835\uDC3F2/3 . Moreover, using the Ornstein-Zernike machinery due to Campanino-Ioffe-Velenik and a result of Ioffe, Shlosman, and Toninelli ruling out pinning of Ising polymers with modified interactions along the boundary, we are able to establish (1) a Brownian excursion scaling limit of a no-wall SOS model with a single level line, and (2) a Ferrari-Spohn scaling limit for an SOS model above a hard wall with a single level. In particular, the 1:2:3 scaling limit of the top level line, if it exists, dominates this Ferrari-Spohn diffusion.In Chapter 3, we consider a random walk model for the SOS level lines introduced by Caputo, Ioffe, and Wachtel. This model consists of a line ensemble of \uD835\uDD25 non-intersecting random walks with a floor at zero and geometrically growing area tilts (thus, the walks are both attracting and repelling). Our main result, joint with M. Hegde and C. Serio, is a 1:2:3 scaling limit on intervals of size AL2/3 and boundary conditions as large as \uD835\uDC342−\uD835\uDF05\uD835\uDC3F1/3 , for any \uD835\uDC34 ≥ (log \uD835\uDC3F)1/3+\uD835\uDEFE , for arbitrarily small \uD835\uDF05,\uD835\uDEFE > 0. Considering such small intervals with high boundary conditions allows us to address several different features of the SOS level lines. As a key input, we obtain Tracy-Widom-type upper tail bounds on the top curves.In Part II, we turn our attention to branching Brownian motion (BBM) and related topics. In Chapters 4-7, we develop a rather complete characterization of the limiting behavior in all dimensions of the extrema of BBM (the particles of largest norm). The extrema of BBM in dimension 1 have been well-understood for over a decade; by contrast, prior to the work in this thesis, no limit theorems were known for the extrema in dimensions 2 and higher. With this understanding in hand, we are able to analyze the front of the BBM particle system: that is, the outer shell of the particle system around the particle of maximal norm. This shell is a random hypersurface that defines the interface between where particles exist and where they do not (since we are looking around the maximal particle). We show a scaling limit, of scaling exponent 3/2, to an explicit random surface. These results are joint with various subsets of J. Berestycki, E. Lubetzky, B. Mallein, and O. Zeitouni.BBM is well-known for its connection to traveling-wave solutions of a reaction-diffusion equation known as the KPP equation. In dimension 1, this connection was exploited to great effect. In Chapter 8, we consider the KPP equation in the multidimensional half-space ℍd, for any \uD835\uDC51 ≥ 2, with Dirichlet boundary conditions. In the study of traveling-wave solutions, the half-space is the first domain considered that is both anisotropic and transversally noncompact. Using a connection with multidimensional BBM killed on the boundary of ℍd, we show that minimal-speed waves are unique up to translation and rotation, but faster waves are not. We relate these traveling waves to 1D traveling waves via shifts and rotations and also identify the waves' asymptotic behavior. The minimal-speed wave in particular possesses a novel feature: far from the boundary, it converges to a logarithmic shift of the 1D minimal-speed wave. These results are joint with J. Berestycki, C. Graham, and B. Mallein.The final chapter of Part II, Chapter 9, is concerned with a universal object in the study of log-correlated fields known as multiplicative chaos. BBM plays a central role in the universality class of log-correlated fields, as a branching structure appears to capture well the effect of log-correlations on extrema. We consider a class of log-correlated fields known as the random Fourier series, or random wave model, with i.i.d. coefficients. When these coefficients are Gaussian, the associated chaos measure is a Gaussian multiplicative chaos (GMC). We show that for general distributions, the associated chaos measure is almost surely mutually absolutely continuous with a coupled GMC measure for a certain range of inverse temperature. This result is joint with X. Kriechbaum.

English

ISBN: 9798286424122Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Branching Brownian motion

The Geometry of Fronts and Interfaces Arising From Interacting Particle Systems /
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520 $a This work is primarily concerned with models of interfaces arising in large systems of interacting particles. Studying these objects will lead us to consider the Solid-on-Solid model, line ensembles of attracting and repelling random walks, branching Brownian motion, traveling-wave solutions to reaction-diffusion equations, the log-correlated random wave model (or random Fourier series), and Gaussian and non-Gaussian multiplicative chaos. This thesis consists of two parts.Part I examines the entropic repulsion phenomena of the (2 + 1)-dimensional Solid-on-Solid (SOS) model at low temperature above a hard wall, with zero boundary conditions. The SOS model is a canonical model for interfaces in equilibrium systems exhibiting phase coexistence. When considered in a box of side-length \uD835\uDC3F, Caputo et al. (2016) established that this random surface rises to a height of size \uD835\uDD25 ≍ log L (in contrast to its no-floor counterpart, which typically lies at O(1) height) via a series of slabs, or "levels," of height 1. The borders of these levels, called level lines, form concentric loops when viewed from above, and these loops admit a deterministic scaling limit when the box is shrunk to unit scale. The inner-most level lines are believed to exhibit a 1:2:3 scaling relation away from the corners of the box: that is, fluctuations of size L1/3 on intervals of length \uD835\uDC3F2/3 away from the limit shape.Our first result, joint with P. Caddeo and E. Lubetzky and presented in Chapter 2, is to establish the scaling relation in the sense of a lower bound: we show that, with high probability, the innermost level line has fluctuations of size at least order \uD835\uDC3F1/3 on intervals of length \uD835\uDC3F2/3 . Moreover, using the Ornstein-Zernike machinery due to Campanino-Ioffe-Velenik and a result of Ioffe, Shlosman, and Toninelli ruling out pinning of Ising polymers with modified interactions along the boundary, we are able to establish (1) a Brownian excursion scaling limit of a no-wall SOS model with a single level line, and (2) a Ferrari-Spohn scaling limit for an SOS model above a hard wall with a single level. In particular, the 1:2:3 scaling limit of the top level line, if it exists, dominates this Ferrari-Spohn diffusion.In Chapter 3, we consider a random walk model for the SOS level lines introduced by Caputo, Ioffe, and Wachtel. This model consists of a line ensemble of \uD835\uDD25 non-intersecting random walks with a floor at zero and geometrically growing area tilts (thus, the walks are both attracting and repelling). Our main result, joint with M. Hegde and C. Serio, is a 1:2:3 scaling limit on intervals of size AL2/3 and boundary conditions as large as \uD835\uDC342−\uD835\uDF05\uD835\uDC3F1/3 , for any \uD835\uDC34 ≥ (log \uD835\uDC3F)1/3+\uD835\uDEFE , for arbitrarily small \uD835\uDF05,\uD835\uDEFE > 0. Considering such small intervals with high boundary conditions allows us to address several different features of the SOS level lines. As a key input, we obtain Tracy-Widom-type upper tail bounds on the top curves.In Part II, we turn our attention to branching Brownian motion (BBM) and related topics. In Chapters 4-7, we develop a rather complete characterization of the limiting behavior in all dimensions of the extrema of BBM (the particles of largest norm). The extrema of BBM in dimension 1 have been well-understood for over a decade; by contrast, prior to the work in this thesis, no limit theorems were known for the extrema in dimensions 2 and higher. With this understanding in hand, we are able to analyze the front of the BBM particle system: that is, the outer shell of the particle system around the particle of maximal norm. This shell is a random hypersurface that defines the interface between where particles exist and where they do not (since we are looking around the maximal particle). We show a scaling limit, of scaling exponent 3/2, to an explicit random surface. These results are joint with various subsets of J. Berestycki, E. Lubetzky, B. Mallein, and O. Zeitouni.BBM is well-known for its connection to traveling-wave solutions of a reaction-diffusion equation known as the KPP equation. In dimension 1, this connection was exploited to great effect. In Chapter 8, we consider the KPP equation in the multidimensional half-space ℍd, for any \uD835\uDC51 ≥ 2, with Dirichlet boundary conditions. In the study of traveling-wave solutions, the half-space is the first domain considered that is both anisotropic and transversally noncompact. Using a connection with multidimensional BBM killed on the boundary of ℍd, we show that minimal-speed waves are unique up to translation and rotation, but faster waves are not. We relate these traveling waves to 1D traveling waves via shifts and rotations and also identify the waves' asymptotic behavior. The minimal-speed wave in particular possesses a novel feature: far from the boundary, it converges to a logarithmic shift of the 1D minimal-speed wave. These results are joint with J. Berestycki, C. Graham, and B. Mallein.The final chapter of Part II, Chapter 9, is concerned with a universal object in the study of log-correlated fields known as multiplicative chaos. BBM plays a central role in the universality class of log-correlated fields, as a branching structure appears to capture well the effect of log-correlations on extrema. We consider a class of log-correlated fields known as the random Fourier series, or random wave model, with i.i.d. coefficients. When these coefficients are Gaussian, the associated chaos measure is a Gaussian multiplicative chaos (GMC). We show that for general distributions, the associated chaos measure is almost surely mutually absolutely continuous with a coupled GMC measure for a certain range of inverse temperature. This result is joint with X. Kriechbaum.
546 $a English
590 $a School code: 0146
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
653 $a Branching Brownian motion
653 $a Gaussian multiplicative chaos
653 $a Interface models
653 $a Log-correlated fields
653 $a Reaction-diffusion equations
653 $a Solid-on-Solid model
690 $a 0405
690 $a 0642
710 2 $a New York University. $b Mathematics. $e degree granting institution. $3 3765635
720 1 $a Lubetzky, Eyal $e degree supervisor.
720 1 $a Zeitouni, Ofer $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-12B.
790 $a 0146
791 $a Ph.D.
792 $a 2025
856 4 0 $u https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31931334