Introduction
Notation and Terminology
Also, for each 𝑖 𝜎𝑖 be the second largest singular value of ∗𝜇𝑖 as an operator on 𝐿2 (⟨supp𝜇𝑖⟩). Let𝑃: 𝐺 ↠ 𝐺/𝐻 be a set map that sends each element of 𝐺 to the corresponding left coset of 𝐻. Most of the time (this will be formalized later) 𝑓 is regular enough that the regularity of 𝑋𝑖,𝑛 implies the regularity of the images 𝑓(𝑋𝑖,𝑛).
Also, for each 𝑖, let us Now associate the second largest singular values of the random walk (𝑄˜𝑗− . 1)∗𝜇𝑖, which we will denote by 𝜎′.
Random Walks
Geometric Proof
For every subgroup𝐻 ≤ 𝐺, this is the space of functions that are constant on every left -coset of 𝐻 (i.e., for𝜈 ∈ M𝐻 and 𝑔1, 𝑔. Note also that if 𝐻 ⊴ 𝐺, then the normal space m𝐻 is canonically isomorphic to 𝐿2(𝐺/𝐻) (by sending𝜈 ∈ M𝐻 to the function send𝑔 𝐻 to𝜈(𝑔 𝐻)), and P𝐻 maps to the set of signed measures on𝐺 of 𝐺/𝐻 the largest value of mass, or phism under this eigenvalue. the convolution operator𝑀𝑖|𝐿2(𝐻𝑗) is 1, which corresponds to the subspace of constant measure on 𝐻𝑖.
The orthogonal complement of a subspace of constant measures on 𝐻𝑗 is 𝐿2(𝐻𝑗) ∩ M0, a subspace of signed measures on 𝐻𝑗 with total mass zero. In particular, (2.1) holds when replacing 𝐻𝑗 with any left side coset of 𝐻𝑗, since multiplication on the right by a random element of 𝐻𝑗 corrects the left cosets. On𝐺, the operator𝑀𝑖 does not shrink the distance between the probability measure and𝜋, but it does shrink the distance between the measure and the subspace M𝐻𝑗.
We bound the distance 𝐿2 between the convolution of 𝜈𝑛 and 𝜋 by bounding the distance 𝐿2 between 𝜈𝑛 and the range{𝜋} = M𝐺 in M. Note that since 𝐻𝑗 is normal, its sets of left and right cosets coincide, so ˜𝜈𝑛 is constant also on each right-hand line of𝐻𝑗.
Algebraic Proof
In the case of any random matrix, 𝑁 is the subset generated by the columns of the random matrix, viewed as elements of Z𝑛. 𝑋𝑚 are 𝜀-balanced random integers with value Z/𝑘Z, where 𝑘 is a multiple of the exponent 𝐺, and write 𝑋 = (𝑋.
Random Groups
Nonabelian Groups
This subsection is devoted to the construction of the universality class for 𝜇𝑢 given by the following theorem:. Assume that ℓ𝑛 are positive integers such that lim𝑛→∞. Then the random group distributions 𝐴𝑛 converge weakly to 𝜇𝑢 as 𝑛→. The argument is analogous to some parts of the proof in [Woo19] that the abelianization of 𝜇𝑢 is universal for abelian groups.
𝑋𝑛+𝑢 are independent random elements of 𝑉 and let 𝐻 = ⟨𝑋. There is a bijection between the surjections𝑉/𝐻 →𝐺and the surjections𝑉 → 𝐺vanishing on𝐻. 3.1). This reduces the problem of computing moments to understanding the image distribution of one random relator at a time. Since the random relators in Theorem 3.1 originate from a random walk, their images in the finite group 𝐺 also originate from a random walk, so they should converge to a uniform distribution.
The following lemma is a more general restatement of [Woo19, Lemma 2.4], but the proof is essentially the same.
Abelian Groups with Dependent Relations
We say that a subset of the data of a random matrix𝑀 with indices𝑆(collectively) is 𝜀-equilibrium if𝜋𝑆(𝑀) is-equilibrium inZ𝑆. The new definition of 𝜀-balanced has a number of desirable properties that help construct new examples of 𝜀-balanced random variables. If the elements of an arbitrary matrix can be divided into independent subsets and each of these subsets of the elements are collectively in equilibrium, then the entire matrix is in equilibrium. The idea is that we can split the columns of the matrix and then the rows so that the resulting parts of the matrix are balanced.
Since 𝑓 is a P-code of distance 𝑤, it remains surjective if we discard all these indices, meaning the images of the 𝑀𝑃. Finally, we can combine all these results to calculate the limiting moments for kernels of (𝑤𝑛, ℎ𝑛, 𝜀𝑛)-balanced random matrices. The bulk of the work for this proof goes into constructing an appropriate complete quotient series to apply Theorem 4.4.
1𝐺 of the 𝑖-step commutator subgroup of𝐺 is central, since by taking the quotient through all 𝑖+1-step commutators, we force𝑖-step commutators to commute with everything in𝐺/𝛾𝑖+.
Partial Results and Future Work
Quotient Sequences
To use Theorem 2.5, we can abstract the condition to make the job easier. 𝐺| in the result of Theorem 2.5, the following definition quite naturally captures the condition for the theorem. If 𝑆is a sublevel𝑖, then 𝑆is also a sublevel of 𝑗 for all 𝑗 > 𝑖. Each group has a unique trivial complete quotient sequence 𝐺 → {𝑒}.
Then there is a quotient sequence. so 𝐻𝑖 is level 𝑗 +𝑖 for each𝑖. According to Remark 4.2, it suffices to show that when 𝑄 is a one-step quotient sequence𝑄. This is possible because surjections maintain normality. be the natural projection and ˜𝑄′. These determine the quotient sequence 𝑄′of length 𝑘. By construction, 𝐻𝑗 is the level of 𝑗 in this quotient sequence. 𝑗, since every set of level 𝑖 is a sublevel of 𝑗 for 𝑗 ≥ 𝑖.
On the other hand, since each 𝐻𝑗 ⊆ ker𝑄. 1, so 𝑄' is a quotient sequence from 𝐺to𝐺. The language of quotient sequences allows a much more concise reformulation of Theorem 2.5.
Equidistribution on Nilpotent Groups
The idea is to replace the random integer vectors in [Woo19] with random vectors in Maltsev coordinates. If 𝑓 is a code with respect to the free generators of 𝑁𝑐,𝑛, then it is a code for every consecutive coefficient𝛾𝑖𝑁𝑐,𝑛/𝛾𝑖+. Each of these switches can include at most 𝑖 distinct elements of 𝑆, so less than 𝑤/𝑖 switches include less than 𝑤 distinct elements of𝑆and #𝑇 < 𝑤.
1𝑊 is free abelian on basic step commutators In particular, the constraint 𝑓|𝛾𝑖𝑊: 𝛾𝑖𝑊 →𝛾𝑖𝐺∩𝐻 is surjective, since every element of 𝑓(𝑊), and thus itself in 𝑓(𝛾𝑖𝑊 ). Finally, we use the prime sequence formalism to prove that a random vector in Maltsev coordinate maps with something close to uniform under a code.
In particular, for every commutator with steps𝑥 in 𝐺, the cyclic subgroup generated by its image in𝐺/𝛾𝑖+.
Future Work
