Brownian intersections, cover times and thick points via trees

There is a close connection between intersections of Brownian motion paths and percolation on trees. Recently, ideas from probability on trees were an important component of the multifractal analysis of Brownian occupation measure, in joint work with A. Dembo, J. Rosen and O. Zeitouni. As a consequence, we proved two conjectures about simple random walk in two dimensions: The first, due to Erd\H{o}s and Taylor (1960), involves the number of visits to the most visited lattice site in the first $n$ steps of the walk. The second, due to Aldous (1989), concerns the number of steps it takes a simple random walk to cover all points of the $n$ by $n$ lattice torus. The goal of the lecture is to relate how methods from probability on trees can be applied to random walks and Brownian motion in Euclidean space

Noise Stability of Weighted Majority

Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of $n$ independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability $\epsilon$, the probability $p_\epsilon$ that the weighted majority changes is at most $C\epsilon^{1/4}$. They asked what is the best possible exponent that could replace 1/4. We prove that the answer is 1/2. The upper bound obtained for $p_\epsilon$ is within a factor of $\sqrt{\pi/2}+o(1)$ from the known lower bound when $\epsilon \to 0$ and $n\epsilon\to \infty$.Comment: six page

Non-amenable products are not treeable

Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product, has infinitely many connected components almost surely.Comment: 8 page

Non-amenable Cayley graphs of high girth have p_c < p_u and mean-field exponents

In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., p_c < p_u. Furthermore, we show that percolation and self-avoiding walk on such graphs have mean-field critical exponents. In particular, the self-avoiding walk has positive speed.Comment: 8 page

Mixing of the upper triangular matrix walk

We study a natural random walk over the upper triangular matrices, with entries in the field $\Z_2$, generated by steps which add row $i+1$ to row $i$. We show that the mixing time of the lazy random walk is $O(n^2)$ which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields $\Z_q$ for $q$ prime.Comment: 11 page

Embeddings of discrete groups and the speed of random walks

For a finitely generated group G and a banach space X let \alpha^*_X(G) (respectively \alpha^#_X(G)) be the supremum over all \alpha\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such that for all x,y\in G we have \|f(x)-f(y)\|\ge c\cdot d_G(x,y)^\alpha. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is \alpha^*(G)=\alpha^*_{L_2}(G) (respectively \alpha^#(G)= \alpha_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then \alpha^#_X(G)\le \frac{1}{p\beta^*(G)}. Here \beta^*(G) is the largest \beta\ge 0 for which there exists a set of generators S of G and c>0 such that for all t\in \N we have \E\big[d_G(W_t,e)\big]\ge ct^\beta, where \{W_t\}_{t=0}^\infty is the canonical simple random walk on the Cayley graph of G determined by S, starting at the identity element. This result is sharp when X=L_p, generalizes a theorem of Guentner and Kaminker and answers a question posed by Tessera. We also show that if \alpha^*(G)\ge 1/2 then \alpha^*(G\bwr \Z)\ge \frac{2\alpha^*(G)}{2\alpha^*(G)+1}. This improves the previous bound due to Stalder and ValetteWe deduce that if we write \Z_{(1)}= \Z and \Z_{(k+1)}\coloneqq \Z_{(k)}\bwr \Z then \alpha^*(\Z_{(k)})=\frac{1}{2-2^{1-k}}, and use this result to answer a question posed by Tessera in on the relation between the Hilbert compression exponent and the isoperimetric profile of the balls in G. We also show that the cyclic lamplighter groups C_2\bwr C_n embed into L_1 with uniformly bounded distortion, answering a question posed by Lee, Naor and Peres. Finally, we use these results to show that edge Markov type need not imply Enflo type.Comment: 24 pages. Added Remark 6.4 and made minor changes in new versio

The critical random graph, with martingales

We give a short proof that the largest component of the random graph $G(n, 1/n)$ is of size approximately $n^{2/3}$. The proof gives explicit bounds for the probability that the ratio is very large or very small.Comment: 13 pages, 1 figure. Revised version. Contains stronger probability deviation bounds and handles the entire scaling window. To appear in Israel Journal of Mathematic

The Threshold for Random k-SAT is 2^k ln2 - O(k)

Let F be a random k-SAT formula on n variables, formed by selecting uniformly and independently m = rn out of all possible k-clauses. It is well-known that if r>2^k ln 2, then the formula F is unsatisfiable with probability that tends to 1 as n tends to infinity. We prove that there exists a sequence t_k = O(k) such that if r < 2^k ln 2 - t_k, then the formula F is satisfiable with probability that tends to 1 as n tends to infinity. Our technique yields an explicit lower bound for the random k-SAT threshold for every k. For k>3 this improves upon all previously known lower bounds. For example, when k=10 our lower bound is 704.94 while the upper bound is 708.94.Comment: Added figures and explained the intuition behind our approach. Made a correction following comments of Chris Calabr

Evolving sets, mixing and heat kernel bounds

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovasz and Kannan, can be refined to apply to the maximum relative deviation $|p^n(x,y)/\pi(y)-1|$ of the distribution at time $n$ from the stationary distribution $\pi$. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.Comment: 21 pages, 4 figures, to appear in PTR

The sharp Hausdorff measure condition for length of projections

In a recent paper, Pertti Mattila asked which gauge functions $\phi$ have the property that for any planar Borel set $A$ with positive Hausdorff measure in gauge $\phi$, the projection of $A$ to almost every line has positive length. We show that integrability near zero of $\phi(r)/(r^2)$, which is known to be sufficient for this property, is also necessary if $\phi$ is regularly varying. Our proof is based on a random construction adapted to the gauge function.Comment: 11 pages, 1 figur