# Probability

# Probability

There is an active and large probability group in the department that has many interactions and collaborations with other groups including statistics, optimization and stochastic modeling. The group encompasses a wide range of research interests and research expertise including, Extreme Value Theory and stochastic dynamical systems with applications in Engineering, Oceanographic, Environmental and Biomedical sciences; Stochastic Analysis, Large Deviations and Stochastic Control, Markov processes and their applications; generalized fiducial inference and applications to biology and engineering; long range dependence and self-similarity; limits of random combinatorial structures including random graphs and random network models, dynamics on and off networks; evolutionary games and applications in mathematical biology, computer science and theoretical machine learning.

The group has grown in breadth as opposed to focusing on a single subdiscipline within probability. The group is involved both in the development of fundamental theory in the various topics mentioned above and in building extensive collaborations with research groups both within in the university as well as with national and international research labs.

## Research Topics

- Probabilistic models for real world networks
- Long range dependence and self-similar phenomena
- Extremal behavior of dynamical systems
- Large deviations and stochastic control
- Randomized algorithms and combinatorial statistics
- Evolutionary game theory
- Stochastic analysis and interacting particle systems
- Probabilistic models in machine learning
- Fiducial inference

## Collaborations

**Within the University:**

Collaboration with UNC-Math to study data assimilation methodologies applied to weather prediction problems as well as network clustering and community detection algorithms. Collaboration with the medical school to use techniques from probabilistic combinatorics to understand brain vascular networks. Collaboration with researchers in the social sciences including political science as well as neuroscience to formulate network models and efficient simulation algorithms. Collaboration with the UNC Pharmacy school on identification of mis-annotated chemical compounds.

**Collaborations with Duke and NCState:**

The group maintains strong ties with the major probability group at Duke and also with researchers at NCState. The triangle probability seminar is nominally organized between UNC and Duke which fosters regular interaction between the two groups as faculty, students and postodocs commute between the two universities to attend the Thursday probability seminar. Graduate students from both Duke and UNC participate regularly in special topic courses in the other university. The Southeastern probability conference has been jointly organized by Duke and UNC since 2012 and results in an annual meeting of researchers in probability in the Southeast. A number of other conferences have been jointly organized between members in these two groups including the Seminar in Stochastic processes, 2013, and Conference on Probability Theory and Combinatorial Optimization, 2015.

**Collaborations Nationally and Internationally:**

DARPA funded research in developing probabilistic and computational framework for model approximation and reduction and uncertainty quantification for multiscale and multiphysics models in material science where rare events play a significant role. Extreme value theory and air pollution regulation and risk analysis including activities such as statistical analyses for the Envirnomental Protection Agency, EPA. Collaboration with US Naval Surface Warfare Center Carderock Division with research including extremes, stochastic dynamics and uncertainty quantification of ship motions. Collaborations with the European Institute of Randomness, Eurandom, for developing mathematical theory related to network models and their implications for real world networks.

## Special Topics Courses

- High-dimensional time series (Pipiras)
- Stochastic Analysis (Budhiraja)
- High Dimensional Probability (Bhamidi)
- Probability for Networks and Algorithms (Fraiman)
- Stability of Markov processes (Banerjee)
- Long Range Dependence (Pipiras)
- Weak Convergence and local weak convergence (Bhamidi)
- Large Deviations and Variational inequalities (Budhiraja)
- Concentration inequalities and Combinatorial Optimization (Bhamidi)
- Variational Problems in Probability and Statistics (Budhiraja)
- Mathematical Models for Complex Networks (Bhamidi)

## Representative Publications

J. Cao & M. Olvera-Cravioto

Connectivity of a general class of inhomogeneous random digraphs

*Random Structures and Algorithms* 56(61):722-774 (2020).

N. Broutin, L. Devroye & N. Fraiman.

Recursive functions on conditional Galton-Watson trees

*Random Structures and Algorithms* 57(2):304–316 (2020).

S. Banerjee & D. Mukherjee

Join-the-Shortest Queue Diffusion Limit in Halfin-Whitt Regime: Tail Asymptotics and Scaling of Extrema

*The Annals of Applied Probability* 29(2):1262-1309 (2019).

V. Belenky, D. Glotzer, V. Pipiras & T. P. Sapsis

Distribution tail structure and extreme value analysis of constrained piecewise linear oscillators

*Probabilistic Engineering Mechanics* 57:1-13 (2019).

G. Didier, M. M. Meerschaert & V. Pipiras

Domain and range symmetries of operator fractional Brownian fields

*Stochastic Processes and their Applications* 128(1):39-78 (2018).

J. Palowitch, S. Bhamidi & A. B. Nobel

Significance-Based Community Detection in Weighted Networks.

*Journal of Machine Learning Research* 18(188):1-48 (2018).

A. Budhiraja & E. Friedlander

Diffusion Approximations for Controlled Weakly Interacting Large Finite State Systems with Simultaneous Jumps

*Annals of Applied Probability* 28(1):204-249 (2018).

N. Fraiman & D. Mitsche

The diameter of Inhomogeneous random graphs

*Random Structures and Algorithms* 53(2):308–326 (2018).

M. Olvera-Cravioto & P. Van der Hoorn

Typical distances in the directed configuration model

*Annals of Applied Probability* 28(3):1739–1792 (2018).

A. Budhiraja, W. Fan & R. Wu

Large Deviations for Brownian Particle Systems with Killing

*Journal of Theoretical Probability* 31:1779–1818 (2018).

S. Bhamidi, J. Jin & A. B. Nobel

Change Point Detection in Network Models: Preferential Attachment and Long Range Dependence

*Annals of Applied Probability* 28(1):35-78 (2018).

S. Banerjee, M. Gordina & P. Mariano

Coupling in the Heisenberg group and its application to gradient estimates

*The Annals of Probability* 46(6):3275-3312 (2018).

S. Bhamidi, P. S. Dey & A. B. Nobel

Energy Landscape for Large Average Submatrix Detection Problems in Gaussian Random Matrices

*Probability Theory and Related Fields* 168:919–983 (2017).

J. D. Wilson, M. J. Denny, S. Bhamidi, S. Cranmer & B. Desmarais

Stochastic Weighted Graphs: Flexible Model Specification and Simulation.

*Social Networks* 49:37-47 (2017).

A. Budhiraja & R. Wu

Some Fluctuation Results for Weakly Interacting Multi-type Particle Systems.

*Stochastic Processes and their Applications*, 126(8):2253-2296 (2016).