Dor Elboim
I am a postdoctoral member at the IAS. Before that I was a PhD student of Allan Sly at Princeton University and a master's student of Ron Peled at Tel Aviv University.
I'm interested in probability theory and applications of probability in mathematical physics and combinatorics.
Contact
dorelboim@gmail.com
Me doing a one-handstand in Times square
Selected papers
Infinite cycles in the interchange process in five dimensions. Submitted.
Joint work with Allan Sly.
Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation. Accepted to Geometric and Functional Analysis (GAFA).
Joint work with Barbara Dembin and Ron Peled.
Minimal surfaces in random environment. Submitted.
Joint work with Barbara Dembin, Daniel Hadas and Ron Peled.
The critical one-dimensional multi-particle DLA. Submitted.
Joint work with Allan Sly and Danny Nam.
All Papers
Minimal surfaces in random environment. Submitted.
Joint work with Barbara Dembin, Daniel Hadas and Ron Peled.
Optimal sample complexity of contrastive learning. ICLR 2024 (spotlight paper).
Joint work with Noga Alon, Dmitrii Avdiukhin, Orr Fischer and Grigory Yaroslavtsev.
Climbing up a random subgraph of the hypercube. Submitted.
Joint work with Michael Anastos, Sahar Diskin and Michael Krivelevich.
On the influence of edges in first-passage percolation. Submitted.
Joint work with Barbara Dembin and Ron Peled.
Infinite cycles in the interchange process in five dimensions. Submitted.
Joint work with Allan Sly. [A talk by Allan]
Heavy and light paths and Hamilton cycles. Information Processing Letters.
Joint work with Sahar Diskin.
Mixing times and cutoff for the TASEP in the high and low density phase. Accepted to Probability and Mathematical Physics.
Joint work with Dominik Schmid.
The Asynchronous DeGroot Dynamics. Submitted.
Joint work with Yuval Peres and Ron Peretz.
Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation. Accepted to Geometric and Functional Analysis (GAFA).
Joint work with Barbara Dembin and Ron Peled. [Talks by Ron, Barbara and me]
On a random model of forgetting. Accepted to The Annals of Applied Probability.
Joint work with Noga Alon and Allan Sly. [A talk by Noga]
Long lines in subsets of large measure in high dimension. Probability Theory and Related Fields (PTRF).
Joint work with Bo'az Klartag. [A talk by Bo'az]
Random necklaces require fewer cuts. Accepted to SIAM Journal on Discrete Mathematics.
Joint work with Noga Alon, Gabor Tardos and Janos Pach. [A talk by Noga]
The critical one-dimensional multi-particle DLA. Submitted.
Joint work with Allan Sly and Danny Nam.
Uniform estimates for almost primes over finite fields. Proceedings of the AMS.
Joint work with Ofir Gorodetsky.
Multiplicative arithmetic functions and the Ewens measure. Accepted to Israel Journal of Mathematics.
Joint work with Ofir Gorodetsky.
Limit distributions for Euclidean random permutations. Communications in Mathematical Physics (CMP).
Joint work with Ron Peled.
Recorded talks and videos
The Interchange process and First Passage Percolation (The IAS short talk).
On the influence of edges in First Passage Percolation.
Infinite cycles in the Interchange process in five dimensions.
Coalescence of geodesics and the BKS midpoint problem in first-passage percolation.
The critical one dimensional multi-particle DLA.
A popular video of me not related to math.
A Mathematical Gallery
Computer simulation of the memory process
The process
Let X(1),X(2),... be a sequence of i.i.d. uniform random variables in [0,1].
In the memory process, the random variables X(1),X(2),... arrive one after the other. The memory list at time n, is a subset S of the variables X(1),...,X(n) defined recursively according to the following rule. When a new variable X(i) arrives, if it is larger than the current minimal element of the list, min(S), then this minimum leaves S and X(i) is added. Otherwise, we just add X(i) to S.
The pictures show the content of the list in four independent samples of the process. In the first two samples we run the process for time 200 and in the last two we run it for time 1000.
Together with Noga and Allan, we show that the content of the memory list is essentially the set of uniform variables that are larger than 1-1/e. In fact we show that the size of the symmetric difference between these two sets is typically of order n^(1/2) and scales to a 3 dimensional Bessel process.
The first picture shows the normalized size of the list over time (up to time 1,000,000). We show that it converges to a Brownian motion.
The second picture shows the evolution over time of the number of elements in the list that are smaller than 1-1/e (in blue) and the number of variables that are larger than 1-1/e and were removed from the list (in red). The third picture shows the number of elements in the list smaller than a+2n^(-1/2), a+n^(-1/2), a, a-n^(-1/2), a-2n^(-1/2) for a=1-1/e and n=10^6. We identify the scaling limits of these processes
A computer simulation of the one dimensional aggregate (MDLA)
The pictures show the size of the aggregate as a function of time in the subcritical and supercritical cases respectively. The process is running for time 300,000 and the different colored graphs are independent samples.
The first picture shows 6 independent samples of the aggregate in the critical case. In this case we show that the aggregate grows at rate t^(2/3).
The second picture shows the speed process corresponding to the blue aggregate (that is, the derivative of the blue curve in the first picture). Together with Allan and Danny, we proved that it converges to a negative power of an (8/3)-dimensional Bessel process.
A computer simulation of the geodesics (shortest paths) in first-passage pecolation.
In this simulation, the edge distribution is uniform in [0,1].
The pictures show two geodesics of length 2000 whose endpoints are at distance of 30. The purple part is the intersection of the two geodesics.
Together with Ron and Barbara, we showed that, under mild assumptions on the edge distribution, geodesics of length n whose endpoints are at distance h<<n^(1/8) will coalesce with high probability.
Minimal surfaces in random environment.
This is a generalization of first passage percolation.
In first passage percolation we study the geodesics in a random environment (random weights). A geodesic is a path such that the sum of weights along the path is minimal. In this work we study the surface (say with zero boundary condition) such that the sum of weights on the surface is minimal.
For a d dimensional minimal surface in a (d+n) dimensional random environment, we study the fluctuations of the surface and how far it is is from being flat. In particular, we want to understand whether the surface is localized (the fluctuations are bounded as the side length of the box tends to infinity) or delocalized (the fluctuations tend to infinity).
Together with Barbara, Daniel and Ron we proved that (for all n) the surfaces are delocalized in dimensions d=1,2,3,4 and localized in dimensions d=5 or higher!
A simulation of a minimal surface with d=1 and n=1.
A simulation of a minimal surface with d=1 and n=2.
A simulation of a minimal surface with d=2 and n=1.
More precisely, for n=1 we obtain the following upper and lower bounds on the fluctuations of the surface as a function of the side length of the box (L).
The cycle structure of random Euclidean permutations
This is a simulation of a random Euclidean permutation on the 2 dimensional torus. The permutation is of size N=50 and the side length L of the torus changes in each picture. Each cycle of the permutation has a different color. Together with Ron, we proved that there is a phase transition in the cycle lengths when the density of points becomes logarithmic. We show that when N/L^2<c log(N), the cycles are finite and when N/L^2>clog(N), the cycles are macroscopic (of order N) and have the Poisson-Dirichlet distribution (for some explicit constant c).