Characterization of cutoff for reversible Markov chains

Transcription

1 Characterization of cutoff for reversible Markov chains Yuval Peres Joint work with Riddhi Basu and Jonathan Hermon 23 Feb 2015 Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 1 / 35

2 Notation Transition matrix - P (reversible). Stationary dist. - π. Reversibility: π(x)p (x, y) = π(y)p (y, x), x, y Ω. Laziness P (x, x) 1/2, x Ω. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 2 / 35

3 TV distance The total-variation distance between 2 dist. µ, ν on Ω is: µ ν TV = d max µ(a) ν(a). A Ω where P t x(y) := P t (x, y). d(t) = d max x Pt x π TV, Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 3 / 35

4 TV distance The total-variation distance between 2 dist. µ, ν on Ω is: µ ν TV = d max µ(a) ν(a). A Ω where P t x(y) := P t (x, y). d(t) = d max x Pt x π TV, The ɛ-mixing-time (0 < ɛ < 1) is: t mix(ɛ) d = min {t : d(t) ɛ} t mix d = t mix(1/4). Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 3 / 35

5 Cutoff - definition Def: a sequence of MCs (X (n) t ) exhibits cutoff if t (n) mix (ɛ) t(n) (1 ɛ) = o(t(n) ), 0 < ɛ < 1/4. (1) mix mix Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 4 / 35

6 Cutoff - definition Def: a sequence of MCs (X (n) t ) exhibits cutoff if t (n) mix (ɛ) t(n) (1 ɛ) = o(t(n) ), 0 < ɛ < 1/4. (1) mix mix ( (w n) is called a cutoff window for (X (n) t ) if: w n = o t (n) mix t (n) mix ), and (ɛ) t(n) (1 ɛ) cɛwn, n 1, ɛ (0, 1/4). mix Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 4 / 35

7 Cutoff Figure : cutoff Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 5 / 35

8 Examples Cutoff was first identified for random transpositions (Diaconis & Shashahani 81). t mix = n log n 2 and w n = n. Many other card shuffling schemes have cutoff. Lazy simple random walk on the hypercube {0, 1} n exhibits a cutoff with t mix = n log n 2 (half the coupon collector time) and w n = n (Aldous 83). Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 6 / 35

9 Examples Cutoff was first identified for random transpositions (Diaconis & Shashahani 81). t mix = n log n 2 and w n = n. Many other card shuffling schemes have cutoff. Lazy simple random walk on the hypercube {0, 1} n exhibits a cutoff with t mix = n log n 2 (half the coupon collector time) and w n = n (Aldous 83). Biased nearest neighbor random walk on an {1, 2,..., n} has cutoff, with a bias toward n has cutoff around E 1[T n]. The glauber dynamics of the Ising model in large temperature has cutoff for many underlying graphs (LS). Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 6 / 35

10 Examples Cutoff was first identified for random transpositions (Diaconis & Shashahani 81). t mix = n log n 2 and w n = n. Many other card shuffling schemes have cutoff. Lazy simple random walk on the hypercube {0, 1} n exhibits a cutoff with t mix = n log n 2 (half the coupon collector time) and w n = n (Aldous 83). Biased nearest neighbor random walk on an {1, 2,..., n} has cutoff, with a bias toward n has cutoff around E 1[T n]. The glauber dynamics of the Ising model in large temperature has cutoff for many underlying graphs (LS). Lazy simple random walk on the n-cycle does not exhibit a cutoff (t mix = Θ(n 2 )). Same for higher dimensional tori of side length n. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 6 / 35

11 Background Many chains are believed to exhibit cutoff. Verifying the occurrence of cutoff rigorously is usually hard. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 7 / 35

12 Background Many chains are believed to exhibit cutoff. Verifying the occurrence of cutoff rigorously is usually hard. The name cutoff was coined by Aldous and Diaconis in their seminal 86 paper: (AD 86) the most interesting open problem : Find verifiable conditions for cutoff. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 7 / 35

13 Background Many chains are believed to exhibit cutoff. Verifying the occurrence of cutoff rigorously is usually hard. The name cutoff was coined by Aldous and Diaconis in their seminal 86 paper: (AD 86) the most interesting open problem : Find verifiable conditions for cutoff. So far cutoff was mostly studied by understanding examples. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 7 / 35

14 Spectral gap & relaxation-time Let λ 2 be the largest non-trivial e.v. of P. Definition: gap = 1 λ 2 - the spectral gap. Def: t rel := 1 - the relaxation-time. gap Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 8 / 35

15 The product condition (Prod. cond.) In a 2004 Aim workshop I proposed that The product condition (Prod. Cond.) - gap (n) t (n) mix (equivalently, t(n) rel = o(t (n) mix )) should imply cutoff for nice reversible chains. (It is a necessary condition for cutoff) Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 9 / 35

16 The product condition (Prod. cond.) In a 2004 Aim workshop I proposed that The product condition (Prod. Cond.) - gap (n) t (n) mix (equivalently, t(n) rel = o(t (n) mix )) should imply cutoff for nice reversible chains. (It is a necessary condition for cutoff) It is not always sufficient (Aldous and Pak). Problem: Find families of MCs s.t. Prod. Cond. = cutoff. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 9 / 35

17 Aldous example Figure : Fixed bias to the right conditioned on a non-lazy step. Different holding probabilities along 2 parallel branches of equal length. t (n) rel = O(1). Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 10 / 35

18 Aldous example d(t) Figure : d(t) is up to negligible terms the probability that the rightmost state was not hit by time t (starting from the leftmost state). Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 11 / 35

19 Hitting and Mixing Def: The hitting time of a set A Ω = T A := min{t : X t A}. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 12 / 35

20 Hitting and Mixing Def: The hitting time of a set A Ω = T A := min{t : X t A}. Hitting times of worst sets are related to mixing: Fact (Random Target): Under laziness and reversibility t mix C max x y π(y)ex[ty] C maxx,y Ex[Ty]. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 12 / 35

21 Hitting and Mixing Def: The hitting time of a set A Ω = T A := min{t : X t A}. Hitting times of worst sets are related to mixing: Fact (Random Target): Under laziness and reversibility t mix C max x y π(y)ex[ty] C maxx,y Ex[Ty]. Generalization (Aldous 82 LW 95) under reversibility and laziness t mix C max x,a π(a)ex[ta]. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 12 / 35

22 Hitting and Mixing Oliveira (2011) and independently P.-Sousi (2011) (+ an by Griffiths et al. (2012)): for any irreducible reversible lazy MC t mix = Θ(t H), where t H := max Ex[TA]. (2) x,a: π(a) 1/2 The threshold 1/2 cannot be improved. For two n-cliques connected by a single edge sets of π measure 1/2 + ɛ are hit in constant number of steps. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 13 / 35

23 Hitting and Mixing Oliveira (2011) and independently P.-Sousi (2011) (+ an by Griffiths et al. (2012)): for any irreducible reversible lazy MC t mix = Θ(t H), where t H := max Ex[TA]. (2) x,a: π(a) 1/2 The threshold 1/2 cannot be improved. For two n-cliques connected by a single edge sets of π measure 1/2 + ɛ are hit in constant number of steps. d(t) 1 P(the other clique was hit by time t) = maybe we should look at tails 2 rather than on expectations. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 13 / 35

24 Hitting and Cutoff Diaconis & Saloff-Coste (06) (separation cutoff) and Ding-Lubetzky-P. (10) (TV cutoff): A seq. of birth and death chains exhibits cutoff iff the Prod. Cond. holds. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 14 / 35

25 Hitting and Cutoff Diaconis & Saloff-Coste (06) (separation cutoff) and Ding-Lubetzky-P. (10) (TV cutoff): A seq. of birth and death chains exhibits cutoff iff the Prod. Cond. holds. Both proofs reduce the problem to establishing concentration of hitting times. We extend their results to weighted nearest-neighbor RWs on trees. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 14 / 35

26 Cutoff for trees Theorem Let (V, P, π) be a lazy Markov chain on a tree T = (V, E). Then t mix(ɛ) t mix(1 ɛ) C log ɛ t rel t mix, for any 0 < ɛ 1/4. In particular, the Prod. Cond. implies cutoff with a cutoff window w n = t (n) rel t(n) mix. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 15 / 35

27 Cutoff for trees Theorem Let (V, P, π) be a lazy Markov chain on a tree T = (V, E). Then t mix(ɛ) t mix(1 ɛ) C log ɛ t rel t mix, for any 0 < ɛ 1/4. In particular, the Prod. Cond. implies cutoff with a cutoff window w n = t (n) rel t(n) mix. DLP (10) - for lazy BD chains t mix(ɛ) t mix(1 ɛ) O(ɛ 1 t rel t mix). ( ) In many cases any cutoff window must be Ω t (n) rel t(n) mix. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 15 / 35

28 A mixing parameter in terms of hitting times Let 0 < ɛ < 1. hit(ɛ) := min{t : P x[t A > t] ɛ : for all x and A Ω s.t. π(a) 1/2}. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 16 / 35

29 Main abstract result Definition: A sequence has hit-cutoff if hit (n) (ɛ) hit (n) (1 ɛ) = o(hit (n) (1/4)) for all 0 < ɛ < 1/4. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 17 / 35

30 Main abstract result Definition: A sequence has hit-cutoff if hit (n) (ɛ) hit (n) (1 ɛ) = o(hit (n) (1/4)) for all 0 < ɛ < 1/4. Main abstract result: Theorem Let (Ω n, P n, π n) be a seq. of finite reversible lazy MCs. Then TFAE: (i) Cutoff. (ii) hit-cutoff. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 17 / 35

31 Main abstract result Definition: A sequence has hit-cutoff if hit (n) (ɛ) hit (n) (1 ɛ) = o(hit (n) (1/4)) for all 0 < ɛ < 1/4. Main abstract result: Theorem Let (Ω n, P n, π n) be a seq. of finite reversible lazy MCs. Then TFAE: (i) Cutoff. (ii) hit-cutoff. We show that (ii) implies the Prod. Cond. (for (i) this is known). Hence (i) (ii) follows from the following quantitative relation between hit and t mix using hit = Θ(t mix): Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 17 / 35

32 To mix - escape and then relax For any reversible irreducible finite lazy chain and any 0 < ɛ 1/4, hit(3ɛ) t rel log(2ɛ) t mix(2ɛ) hit(ɛ) + t rel log(2ɛ) Terms involving t rel are negligible under the Prod. Cond.. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 18 / 35

33 Hitting times when X 0 π Easy direction: to mix, the chain must first escape from small sets = first stage of mixing. Fact: Let A Ω be such that π(a) 1/2. Then (under reversibility) P π[t A > 2st rel ] e s, for all s 0. 2 Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 19 / 35

34 Hitting times when X 0 π Easy direction: to mix, the chain must first escape from small sets = first stage of mixing. Fact: Let A Ω be such that π(a) 1/2. Then (under reversibility) P π[t A > 2st rel ] e s, for all s 0. 2 Follows by the spectral decomposition of P A c (the chain killed upon hitting A). Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 19 / 35

35 Hitting times when X 0 π Easy direction: to mix, the chain must first escape from small sets = first stage of mixing. Fact: Let A Ω be such that π(a) 1/2. Then (under reversibility) P π[t A > 2st rel ] e s, for all s 0. 2 Follows by the spectral decomposition of P A c (the chain killed upon hitting A). By a coupling argument, P x[t A > t + 2st rel ] d(t) + P π[t A > 2st rel ]. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 19 / 35

36 The hard direction We say the chain is ɛ-mixed w.r.t. A Ω at time t + s if for every x P t+s x (A) π(a) ɛ. We say that the event T G t serves as a certificate of being ɛ-mixed w.r.t. A Ω at time t + s if P x[x t+s A T G t] π(a) ɛ. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 20 / 35

37 Application of the max ineq. Claim: Denote t := hit(ɛ), s := t rel log(2/ɛ). Then t mix(2ɛ) t + s. Proof: want for every A Ω to have G = G s(a) Ω s.t. π(g) 1/2 and T G t serves as a certificate of being ɛ-mixed w.r.t. A at time t + s. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 21 / 35

38 Application of the max ineq. Claim: Denote t := hit(ɛ), s := t rel log(2/ɛ). Then t mix(2ɛ) t + s. Proof: want for every A Ω to have G = G s(a) Ω s.t. π(g) 1/2 and T G t serves as a certificate of being ɛ-mixed w.r.t. A at time t + s. Consider G = G s(a) := { } g : s s, P s g(a) π(a) 2e s/t rel = ɛ. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 21 / 35

39 Application of the max ineq. Claim: Denote t := hit(ɛ), s := t rel log(2/ɛ). Then t mix(2ɛ) t + s. Proof: want for every A Ω to have G = G s(a) Ω s.t. π(g) 1/2 and T G t serves as a certificate of being ɛ-mixed w.r.t. A at time t + s. Consider G = G s(a) := { } g : s s, P s g(a) π(a) 2e s/t rel = ɛ. A consequence of Starr s maximal ineq. is that π(g) 1/2. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 21 / 35

40 Application of the max ineq. Claim: Denote t := hit(ɛ), s := t rel log(2/ɛ). Then t mix(2ɛ) t + s. Proof: want for every A Ω to have G = G s(a) Ω s.t. π(g) 1/2 and T G t serves as a certificate of being ɛ-mixed w.r.t. A at time t + s. Consider G = G s(a) := { } g : s s, P s g(a) π(a) 2e s/t rel = ɛ. A consequence of Starr s maximal ineq. is that π(g) 1/2. = P x[t G > t] ɛ. (by def. of t). For any x, A: P t+s x (A) π(a) P x[t G > t] + max g G, s s P s g(a) π(a) 2ɛ. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 21 / 35

41 Remark: The threshold 1/2 in the definition of hit can be replaced by any 0 < α < 1, i.e. at a price of an additive error of t rel log(1 α) the analysis extends to hit α,x(ɛ) := min{t : P x[t A > t] ɛ : for all A Ω s.t. π(a) α}, hit α(ɛ) := max x Ω hitα,x(ɛ) If α > 1/2 one needs to assume that the Prod. Cond. holds to maintain the equivalence of the 2 notions of cutoff. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 22 / 35

42 Remark: The threshold 1/2 in the definition of hit can be replaced by any 0 < α < 1, i.e. at a price of an additive error of t rel log(1 α) the analysis extends to hit α,x(ɛ) := min{t : P x[t A > t] ɛ : for all A Ω s.t. π(a) α}, hit α(ɛ) := max x Ω hitα,x(ɛ) If α > 1/2 one needs to assume that the Prod. Cond. holds to maintain the equivalence of the 2 notions of cutoff. = When t rel is known, it suffices to consider escape probabilities from small sets in order to bound the mixing-time. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 22 / 35

43 Tools Def: For f R Ω, t 0, define P t f R Ω by P t f(x) := E x[f(x t)] = P t (x, y)f(y). y Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 23 / 35

44 Tools Def: For f R Ω, t 0, define P t f R Ω by P t f(x) := E x[f(x t)] = P t (x, y)f(y). y For f R Ω define E π[f] := x Ω π(x)f(x) and f 2 2 := E π[f 2 ]. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 23 / 35

45 Tools Def: For f R Ω, t 0, define P t f R Ω by P t f(x) := E x[f(x t)] = P t (x, y)f(y). y For f R Ω define E π[f] := x Ω π(x)f(x) and f 2 2 := E π[f 2 ]. For g R Ω denote Var πg := g E π[g] 2 2. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 23 / 35

46 Tools Def: For f R Ω, t 0, define P t f R Ω by P t f(x) := E x[f(x t)] = P t (x, y)f(y). y For f R Ω define E π[f] := x Ω π(x)f(x) and f 2 2 := E π[f 2 ]. For g R Ω denote Var πg := g E π[g] 2 2. The following is well-known and follows from elementary linear-algebra. Lemma (Contraction Lemma) Let (Ω, P, π) be a finite rev. irr. lazy MC. Let A Ω. Let t 0. Then Var πp t 1 A e 2t/t rel /4. (3) Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 23 / 35

47 Maximal Inequality The main ingredient in our approach is Starr s maximal-inequality (66) (refines Stein s max-inequality (61)) Theorem (Maximal inequality) Let (Ω, P, π) be a lazy irreducible reversible Markov chain. Let f R Ω. Define the corresponding maximal function f R Ω as f (x) := sup P k (f)(x) = sup E x[f(x k )]. 0 k< 0 k< Then for 1 < p <, f p q f p 1/p + 1/q = 1. (4) Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 24 / 35

48 Maximal Inequality The main ingredient in our approach is Starr s maximal-inequality (66) (refines Stein s max-inequality (61)) Theorem (Maximal inequality) Let (Ω, P, π) be a lazy irreducible reversible Markov chain. Let f R Ω. Define the corresponding maximal function f R Ω as f (x) := sup P k (f)(x) = sup E x[f(x k )]. 0 k< 0 k< Then for 1 < p <, f p q f p 1/p + 1/q = 1. (4) Applying this (with p = 2) to f s := P s (1 A π(a)) in conjunction with the Contraction Lemma yields the desired bound on the size of G c {(f s ) 2 > 8 f s 2 2} {(f s ) 2 > 2 f s 2 2}. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 24 / 35

49 Trees Let: T := (V, E) be a finite tree. (V, P, π) a lazy MC corresponding to some (lazy) weighted nearest-neighbor walk on T (i.e. P (x, y) > 0 iff {x, y} E or y = x). Fact: (Kolmogorov s cycle condition) every MC on a tree is reversible. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 25 / 35

50 Trees Can the tree structure be used to determine the identity of the worst sets? Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 26 / 35

51 Trees Can the tree structure be used to determine the identity of the worst sets? Easier question: what set of π measure 1/2 is the hardest to hit in a birth & death chain with state space [n] := {1, 2,..., n}? Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 26 / 35

52 Trees Can the tree structure be used to determine the identity of the worst sets? Easier question: what set of π measure 1/2 is the hardest to hit in a birth & death chain with state space [n] := {1, 2,..., n}? Answer: take a state m with π([m]) 1/2 and π([m 1]) < 1/2. Then the set worst set would be either [m] or [n] \ [m 1]. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 26 / 35

53 Trees How to generalize this to trees? Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 27 / 35

54 Central vertex C2 o (Ci) 1/2 for all i C1 C3 C4 Figure : A vertex o V is called a central-vertex if each connected component of T \ {o} has stationary probability at most 1/2. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 28 / 35

55 Trees There is always a central-vertex (and at most 2). We fix one, denote it by o and call it the root. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 29 / 35

56 Trees There is always a central-vertex (and at most 2). We fix one, denote it by o and call it the root. There is a simple reduction: T o is concentrated (from worst leaf) = hit-cutoff. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 29 / 35

57 Trees It suffices to show that E x[t o] = Ω(t mix) = T o is concentrated if X 0 = x. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 30 / 35

58 Trees It suffices to show that E x[t o] = Ω(t mix) = T o is concentrated if X 0 = x. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 30 / 35

59 Trees o=vk vk-1 v3 v2 v1 x=v0 Figure : Let v 0 = x,v 1,..., v k = o be the vertices along the path from x to o. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 31 / 35

60 Trees Proof of Concentration: Var x[t o] Ct rel t mix: It suffices to show that Var x[t o] 4t rel E x[t o]. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 32 / 35

61 Trees Proof of Concentration: Var x[t o] Ct rel t mix: It suffices to show that Var x[t o] 4t rel E x[t o]. If X 0 = x then T o is the sum of crossing times of the edges along the path between x: τ i := T vi T vi 1 d = T vi underx 0 = v i 1 Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 32 / 35

62 Trees Proof of Concentration: Var x[t o] Ct rel t mix: It suffices to show that Var x[t o] 4t rel E x[t o]. If X 0 = x then T o is the sum of crossing times of the edges along the path between x: τ i := T vi T vi 1 d = T vi underx 0 = v i 1 τ 1,..., τ k are independent = it suffices to bound the sum of their variances Var x[t o] = Var x[τ i]. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 32 / 35

63 Trees Proof of Concentration: Var x[t o] Ct rel t mix: It suffices to show that Var x[t o] 4t rel E x[t o]. If X 0 = x then T o is the sum of crossing times of the edges along the path between x: τ i := T vi T vi 1 d = T vi underx 0 = v i 1 τ 1,..., τ k are independent = it suffices to bound the sum of their variances Var x[t o] = Var x[τ i]. The law of each τ i is a mixture of geometrics with parameters 1/2t rel = Varx[τ i] 4t rel E x[τ i] = 4t rel E x[t o]. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 32 / 35

64 Beyond trees The tree assumption can be relaxed. We can treat jumps to vertices of bounded distance on a tree (i.e. the length of the path from u to v in the tree (which is now just an auxiliary structure) is > r = P (u, v) = 0) under some mild necessary assumption. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 33 / 35

65 Beyond trees The tree assumption can be relaxed. We can treat jumps to vertices of bounded distance on a tree (i.e. the length of the path from u to v in the tree (which is now just an auxiliary structure) is > r = P (u, v) = 0) under some mild necessary assumption. Previously the BD assumption could not be relaxed mainly due to it being exploited via a representation of hitting times result for BD chains. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 33 / 35

66 Beyond trees The tree assumption can be relaxed. We can treat jumps to vertices of bounded distance on a tree (i.e. the length of the path from u to v in the tree (which is now just an auxiliary structure) is > r = P (u, v) = 0) under some mild necessary assumption. Previously the BD assumption could not be relaxed mainly due to it being exploited via a representation of hitting times result for BD chains. In particular, if P (u, v) δ > 0 for all u, v s.t. d T (u, v) r (and otherwise P (u, v) = 0), then cutoff the Prod. Cond. holds. Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 33 / 35

67 Open problems What can be said about the geometry of the worst sets in some interesting particular cases (say, transitivity or monotonicity)? Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 34 / 35

68 Open problems What can be said about the geometry of the worst sets in some interesting particular cases (say, transitivity or monotonicity)? When can the worst sets be described as { f 2 C} (P f 2 = λ 2f 2)? Is it true that for transitive graphs the mixing time is bounded by the expected time it takes the walk to escape a ball containing at least half the vertices? Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 34 / 35

69 Open problems What can be said about the geometry of the worst sets in some interesting particular cases (say, transitivity or monotonicity)? When can the worst sets be described as { f 2 C} (P f 2 = λ 2f 2)? Is it true that for transitive graphs the mixing time is bounded by the expected time it takes the walk to escape a ball containing at least half the vertices? Does the product condition imply cutoff for transitive graphs under certain mild conditions? Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 34 / 35

70 Open problems What can be said about the geometry of the worst sets in some interesting particular cases (say, transitivity or monotonicity)? When can the worst sets be described as { f 2 C} (P f 2 = λ 2f 2)? Is it true that for transitive graphs the mixing time is bounded by the expected time it takes the walk to escape a ball containing at least half the vertices? Does the product condition imply cutoff for transitive graphs under certain mild conditions? Does contraction implies cutoff? (under some mild conditions...) Joint work with Riddhi Basu and Jonathan Hermon Characterization of cutoff for reversible Markov chains 34 / 35