Péter P. Varjú’s research while affiliated with University of Cambridge and other places

... One of the most challenging problems in Fractal Geometry is to understand how a stationary measure distributes mass when the underlying iterated function system is overlapping. The exact overlaps conjecture and the study of Bernoulli convolutions are two particular instances of this problem (see [28,29,43,44,46,50,51] and the references therein). In this paper, we show that for self-conformal and self-similar measures, it is possible to disintegrate these measures over a family of measures for which we have lots of control over how mass is distributed. ...

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Disintegration results for fractal measures and applications to Diophantine approximation

... As a notable recent example, in P. Shmerkin's recent proof of the Furstenberg's intersection conjecture [Shm19], he computes the L q -spectrum of a large class of dynamically self-similar measures and relates such results to the multifractal analysis of slices of sets. This information about L q -spectra also implies L p -smoothness properties in the question of absolute continuity of Bernoulli convolutions (see [Var18] for some background on this classic problem). For more detail on the geometry of measures and multifractal analysis, we refer the reader to the foundational work by Olsen [Ols95] and the classic texts of Falconer [Fal97] and Pesin [Pes98]. ...

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Multifractal Analysis of Measures Arising from Random Substitutions

... It will illustrate that the random mapping technique can be effective in constructing strong stationary times in situations where they are difficult to find and have lead to numerous mistakes in the past. While there is room for improvement in our estimates, we hope this new approach will help the understanding of the convergence to equilibrium of related random walks, see for instance Hermon and Thomas [11], Breuillard and Varjú [2], Eberhard and Varjú [9] or Chatterjee and Diaconis [5] for very recent progress in this direction. the matrices of the form where x k,l Z M for 1 ¤ k l ¤ N , the group operation corresponds to the matrix multiplication. ...

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Strong Stationary Times for Finite Heisenberg Walks

... ) We end this introduction with a few remarks about the implications Theorem 2 has on the Fourier decay problem for fractal measures. Recall that a probability measure ν on R is called a Rajchman measure if lim |q|→∞ ν(q) = 0. Combining recent breakthroughs on the Fourier decay problem [13,25,20,4,3,1,5] we know that if a C ω (R) self-conformal IFS admits a non-Rajchman self-conformal measure then: The IFS must be self-similar; It must have contraction ratios that are all powers of some r ∈ (0, 1) such that r −1 is a Pisot number; And, up to affine conjugation, all translates are in Q(r ). Now, it is natural to ask about analogues of this result in C r (R)-regularity, 1 ≤ r ≤ ∞ -see e.g. ...

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How linear can a non-linear hyperbolic IFS be?

... In this work we establish cutoff for random walks on randomly generated graphs at an entropic time. There have been multiple recent works proving cutoff at an entropic time, including [4,6,8,9,11,12,15,17,18]. For a more detailed overview please refer to [18]. ...

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Phase transition for random walks on graphs with added weighted random matching

... In another, more algebraic direction, one may ask what typical Galois-theoretic properties of the polynomials in are. For fixed N of cardinality at least 2, a folklore conjecture [11,9,6,4,3] asserts that a random polynomial ∈ of degree has the symmetric group as Galois group over Q with probability tending to 1 as tends to infinity. It is an outstanding challenge to rule out the alternating group as likely Galois group. ...

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Topology of zero sets of polynomials with square discriminant

... Let us finally mention the result of [27], which shows that the expansion of a simple random walk on a semi-direct product (Z pZ) d ⋊ SL d (Z pZ), p prime, and that of the projection to SL d (Z pZ), are of the same order. Hence in this case the mixing properties of an affine random walk are essentially determined by the product of random matrices. ...

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Accelerating abelian random walks with hyperbolic dynamics

... The first examples of IFSs with this 'super-exponential concentration' property were given in [Bak;BK3], and there has been further work on the dimension theory of such IFSs [Rap; RV]. There are also longstanding open problems about dimensions and absolute continuity of overlapping self-similar measures such as Bernoulli convolutions [BV;Erd1;Erd2;Var1;Var2;Var3]. ...

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Interpolating between Hausdorff and box dimension

... The study of Bernoulli convolutions has a long history, going back to Jessen and Wintner [30] that proved that the distribution ν c is either singular or absolutely continuous with respect to the Lebesgue measure (see e.g. Varjú [44] or Peres et al. [38] for recent surveys of the topic and Benaim and El Karoui [8], chapter 1, for a basic introduction). In our setting, the following conclusions can be drawn from the existing literature: ...

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Opinion dynamics in communities with major influencers and implicit social influence via mean-field approximation

... Since the B n 's are bounded and 1 − µ ∈ (0, 1), the partial sums µ n k=0 (1 − µ) k B k converge almost surely, thus Z ∞ is well defined. Moreover, if B is an independent copy of the B n 's, then We remark that in this case, the infinite (random) series n (1 − µ) n B n = Z ∞ /µ is termed the Bernoulli convolution, which has been studied extensively [18,34,38,39,41]. The following characterization of ρ ∞ is well known in this literature; here we provide a different proof using the PDE (3.4): ...

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Fractal opinions among interacting agents