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1
Intro
2
Interacting Stochastic Processes
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A Prototype Examples Pairwise Interacting Diffusions
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Global Empirical Measure Process
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Key Questions
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Outline of the Rest of the Talk
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Classical Mean-Field Results for Interacting Diffusions
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Summary of the Classical Case
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Challenges in the Sparse Regime
10
Local weak convergence of graphs
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Local convergence of marked graphs
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Examples of local weak convergence of deterministic graphs
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Modes of Local Convergence for Random Graph Sequences
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Other Examples of Local weak convergence of random graphs
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A More General Class of Interacting Diffusions
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1. Process Convergence Results
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Global Empirical Measure Convergence Results
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2. Global Empirical Measure Convergence
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Marginal Dynamics on the Line
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Key Properties of the Marginal Dynamics/Local Equations
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Elements of the Proof: 1. A Filtering Lemma
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Elements of the Proof: 2. A Markov Random Field Property
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Summary: Beyond Mean-Field Limits
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Infinite d-regular trees
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Unimodular Galton-Watson trees
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Marginal Dynamics on Galton Watson Trees
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Interacting Jump Process Dynamics
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Analogous Convergence Results Assumption
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Convergence Results for Jump Processes (contd.)
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Marginal Dynamics for Jump Processes on A-Regular Trees
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Markovian Approximations to the Local Equations
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Detecting Phase Transitions via Markov Approximations
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Markovian Approximations for Transient Behavior
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Acknowledgment for Numerical Simulations
Description:
Explore a 50-minute lecture on interacting stochastic processes on sparse random graphs, delivered by Kavita Ramanan for the International Mathematical Union. Delve into recent progress in characterizing hydrodynamic limits and marginal dynamics of stochastic processes on sparse random interaction graphs. Compare their behavior with classically studied mean-field limits arising from complete interaction graphs. Examine applications in statistical physics, neuroscience, biology, and engineering. Learn about local weak convergence of graphs, process convergence results, and global empirical measure convergence. Investigate marginal dynamics on various graph structures, including infinite d-regular trees and unimodular Galton-Watson trees. Discover analogous convergence results for jump processes and explore Markovian approximations for detecting phase transitions and analyzing transient behavior.

Kavita Ramanan - Interacting Stochastic Processes on Sparse Random Graphs

International Mathematical Union
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#Science #Physics #Statistical Physics #Phase Transitions
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Intro