1. A Local Central Limit Theorem for the
Number of Triangles in a Random Graph
Justin Gilmer
Rutgers University
Department of Mathematics
(Joint with Swastik Kopparty)
2. Publications
● “On the Density of Happy Numbers”, Integers, 2013
● “Composition Limits and Separating Examples for some
Boolean Function Complexity Measures”, Combinatorica,
(To Appear). With Michael Saks and Shrikanth Shrinivasan
●
“A New Approach to the Sensitivity Conjecture”, 6th
Innovations in Theoretical Computer Science Conference,
Jan 2015. With Michael Saks and Michal Koucky
● “A Local Central Limit Theorem for the Number of
Triangles in a Random Graph”, Random Structures and
Algorithms, (Under Review). With Swastik Kopparty
3. Graphs and Triangles
● A graph is an ordered pair G = (V,E) of vertices
V and edges E.
|V| = 10, |E| = 12
5 triangles
4. Erdos-Renyi Random Graph Model
● G(n,p) is a random graph on n vertices, each edge
appears independently with probability p.
● “On Random Graphs” (1959), 5519 citations.
G(4,1/2)
7. Central Limit Theorem for S_n
Central Limit Theorem [ER60]: For any fixed a,b
● Similar results hold for subgraph counts, for any
fixed graph [Rucinski 88].
● The joint distribution of subgraph counts for fixed
graphs K_1,...,K_m is a multivariate normal
distribution [Janson 92].
8. Central Limit Theorem for S_n
Proof (Method of Moments):
Central Limit Theorem [ER60]: For any fixed a,b
9. CLT vs Local CLT
Question: What if we want to estimate Pr[S_n = k]?
Local CLT [Gilmer, Kopparty 2014]:
10. History of the Local CLT
Theorem [de Moivre – Laplace 1738]:
11. Recent Results
● What if X_i aren't Bernoulli?
● What if X_i aren't identically distributed?
● Improve error bounds?
● Local CLTs for markov chains...
What if X_i aren't independent??
