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BEGIN:VEVENT
SUMMARY:Bhargav Narayanan (Rutgers)
DTSTART:20200414T130000Z
DTEND:20200414T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/1
DESCRIPTION:Title: Thresholds\nby Bhargav Narayanan (Rutgers) as part of Oxfo
rd discrete mathematics and probability seminar\n\n\nAbstract\nI'll discus
s our recent proof of a conjecture of Talagrand\, a fractional version of
the "expectation-threshold" conjecture of Kahn and Kalai. As a consequence
of this result\, we resolve various (heretofore) difficult problems in pr
obabilistic combinatorics and statistical physics.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ron Peled (Tel Aviv)
DTSTART:20200414T143000Z
DTEND:20200414T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/2
DESCRIPTION:Title: Site percolation on planar graphs and circle packings\nby
Ron Peled (Tel Aviv) as part of Oxford discrete mathematics and probabilit
y seminar\n\n\nAbstract\nColor each vertex of an infinite graph blue with
probability p and red with probability 1-p\, independently among vertices.
For which values of p is there an infinite connected component of blue ve
rtices? The talk will focus on this classical percolation problem for the
class of planar graphs. Recently\, Itai Benjamini made several conjectures
in this context\, relating the percolation problem to the behavior of sim
ple random walk on the graph. We will explain how partial answers to Benja
mini's conjectures may be obtained using the theory of circle packings. Am
ong the results is the fact that the critical percolation probability admi
ts a universal lower bound for the class of recurrent plane triangulations
. No previous knowledge on percolation or circle packings will be assumed.
\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Agelos Georgakopoulos (Warwick)
DTSTART:20200421T130000Z
DTEND:20200421T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/3
DESCRIPTION:Title: The percolation density $\\theta(p)$ is analytic\nby Agelo
s Georgakopoulos (Warwick) as part of Oxford discrete mathematics and prob
ability seminar\n\n\nAbstract\nWe prove that for Bernoulli bond percolatio
n on ℤd\, d≥2\, the percolation density θ(p) (defined as the probabil
ity of the origin lying in an infinite cluster) is an analytic function of
the parameter in the supercritical interval (p_c\,1]. This answers a ques
tion of Kesten from 1981.\n The proof involves a little bit of elementa
ry complex analysis (Weierstrass M-test)\, a few well-known results from p
ercolation theory (Aizenman-Barsky/Menshikov theorem)\, but above all comb
inatorial ideas. We used a new notion of contours\, bounds on the number o
f partitions of an integer\, and the inclusion-exclusion principle\, to ob
tain a refinement of a classical argument of Peierls that settled the 2-di
mensional case in 2018. More recently\, we coupled these techniques with a
renormalisation argument to handle all dimensions.\n Joint work with C
hristoforos Panagiotis.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cristina Toninelli (Paris Dauiphine)
DTSTART:20200421T143000Z
DTEND:20200421T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/4
DESCRIPTION:Title: Bootstrap percolation and kinetically constrained spin models:
critical time scales\nby Cristina Toninelli (Paris Dauiphine) as part
of Oxford discrete mathematics and probability seminar\n\n\nAbstract\nRec
ent years have seen a great deal of progress in understanding the behavior
of bootstrap percolation models\, a particular class of monotone cellular
automata. In the two dimensional lattice there is now a quite complete un
derstanding of their evolution starting from a random initial condition\,
with a universality picture for their critical behavior. Here we will cons
ider their non-monotone stochastic counterpart\, namely kinetically constr
ained models (KCM). In KCM each vertex is resampled (independently) at rat
e one by tossing a p-coin iff it can be infected in the next step by the b
ootstrap model. In particular infection can also heal\, hence the non-mono
tonicity. Besides the connection with bootstrap percolation\, KCM have an
interest in their own : when p shrinks to 0 they display some of the most
striking features of the liquid/glass transition\, a major and still large
ly open problem in condensed matter physics.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grégory Miermont (ENS Lyon)
DTSTART:20200428T130000Z
DTEND:20200428T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/5
DESCRIPTION:Title: The breadth-first construction of scaling limits of graphs wit
h finite excess\nby Grégory Miermont (ENS Lyon) as part of Oxford dis
crete mathematics and probability seminar\n\n\nAbstract\nRandom graphs wit
h finite excess appear naturally in at least two different settings: rando
m graphs in the critical window (aka critical percolation on regular and o
ther classes of graphs)\, and unicellular maps of fixed genus. In the firs
t situation\, the scaling limit of such random graphs was obtained by Adda
rio-Berry\, Broutin and Goldschmidt based on a depth-first exploration of
the graph and on the coding of the resulting forest by random walks. This
idea originated in Aldous' work on the critical random graph\, using inste
ad a breadth-first search approach that seem less adapted to taking graph
scaling limits. We show hat this can be done nevertheless\, resulting in s
ome new identities for quantities like the radius and the two-point functi
on of the scaling limit. We also obtain a similar "breadth-first" construc
tion of the scaling limit of unicellular maps of fixed genus. This is base
d on joint work with Sanchayan Sen.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Bernardi (Brandeis)
DTSTART:20200428T143000Z
DTEND:20200428T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/6
DESCRIPTION:Title: Percolation on triangulations\, and a bijective path to Liouvi
lle quantum gravity\nby Olivier Bernardi (Brandeis) as part of Oxford
discrete mathematics and probability seminar\n\n\nAbstract\nAbstract: I wi
ll discuss the percolation model on planar triangulations\, and present a
bijection that is key to relating this model to some fundamental probabili
stic objects. I will attempt to achieve several goals:\n1. Present the sit
e-percolation model on random planar triangulations.\n2. Provide an inform
al introduction to several probabilistic objects: the Gaussian free field\
, Schramm-Loewner evolutions\, and the Brownian map.\n3. Present a bijecti
ve encoding of percolated triangulations by certain lattice paths\, and ex
plain its role in establishing exact relations between the above-mentioned
objects.\nThis is joint work with Nina Holden\, and Xin Sun.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liana Yepremyan (LSE)
DTSTART:20200505T130000Z
DTEND:20200505T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/7
DESCRIPTION:Title: Ryser's conjecture and more\nby Liana Yepremyan (LSE) as p
art of Oxford discrete mathematics and probability seminar\n\nAbstract: TB
A\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benny Sudakov (ETH Zurich)
DTSTART:20200505T143000Z
DTEND:20200505T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/8
DESCRIPTION:Title: Multidimensional Erdős-Szekeres theorem\nby Benny Sudakov
(ETH Zurich) as part of Oxford discrete mathematics and probability semin
ar\n\n\nAbstract\nAbstract: The classical Erdős-Szekeres theorem dating b
ack almost a hundred years states that any sequence of $(n-1)2+1$ distinct
real numbers contains a monotone subsequence of length n. This theorem ha
s been generalised to higher dimensions in a variety of ways but perhaps t
he most natural one was proposed by Fishburn and Graham more than 25 years
ago. They raise the problem of how large should a d-dimesional array be i
n order to guarantee a "monotone" subarray of size $n \\times n \\times \\
ldots \\times n$. In this talk we discuss this problem and show how to imp
rove their original Ackerman-type bounds to at most a triple exponential.
(Joint work with M. Bucic and T. Tran)\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tamar Ziegler (Hebrew University Jerusalem)
DTSTART:20200512T130000Z
DTEND:20200512T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/9
DESCRIPTION:Title: Sections of high rank varieties and applications\nby Tamar
Ziegler (Hebrew University Jerusalem) as part of Oxford discrete mathemat
ics and probability seminar\n\n\nAbstract\nI will describe some recent wor
k with D. Kazhdan where we obtain results in algebraic geometry\, inspired
by questions in additive combinatorics\, via analysis over finite fields.
Specifically we are interested in quantitative properties of polynomial r
ings that are independent of the number of variables. A sample application
is the following theorem : Let $V$ be a complex vector space\, $P$ a high
rank polynomial of degree $d$\, and $X$ the null set of $P\, X=\\{v|P(v)=
0\\}$. Any function $f:X\\to C$ which is polynomial of degree $d$ on lines
in $X$ is the restriction of a degree $d$ polynomial on $V$.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anand Pillay (Notre Dame)
DTSTART:20200512T143000Z
DTEND:20200512T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/10
DESCRIPTION:Title: Approximate subgroups with bounded VC dimension\nby Anand
Pillay (Notre Dame) as part of Oxford discrete mathematics and probabilit
y seminar\n\n\nAbstract\nThis is joint with Gabe Conant. We give a structu
re theorem for finite subsets A of arbitrary groups G such that A has "sma
ll tripling" and "bounded VC dimension". Roughly\, A will be a union of a
bounded number of translates of a coset nilprogession of bounded rank and
step (up to a small error).\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gal Kronenberg (Oxford)
DTSTART:20200519T130000Z
DTEND:20200519T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/11
DESCRIPTION:Title: The maximum length of $K_r$-Bootstrap Percolation\nby Gal
Kronenberg (Oxford) as part of Oxford discrete mathematics and probabilit
y seminar\n\n\nAbstract\nHow long does it take for a pandemic to stop spre
ading? When modelling an infection process\, especially these days\, this
is one of the main questions that comes to mind. In this talk\, we conside
r this question in the bootstrap percolation setting.\nGraph-bootstrap per
colation\, also known as weak saturation\, was introduced by Bollobás in
1968. In this process\, we start with initial "infected" set of edges E0\,
and we infect new edges according to a predetermined rule. Given a graph
H and a set of previously infected edges E_t ⊆ E(Kn)\, we infect a non-i
nfected edge e if it completes a new copy of H in G=([n] \, Et ∪ {e}). A
question raised by Bollobás asks for the maximum time the process can ru
n before it stabilizes. Bollobás\, Przykucki\, Riordan\, and Sahasrabudhe
considered this problem for the most natural case where H=Kr. They answer
ed the question for r ≤ 4 and gave a non-trivial lower bound for every r
≥ 5. They also conjectured that the maximal running time is o(n2) for e
very integer r. We disprove their conjecture for every r ≥ 6 and we give
a better lower bound for the case r=5\; in the proof we use the Behrend c
onstruction. This is a joint work with József Balogh\, Alexey Pokrovskiy\
, and Tibor Szabó.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eyal Lubetzky (Courant)
DTSTART:20200519T143000Z
DTEND:20200519T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/12
DESCRIPTION:Title: Maximum height of 3D Ising interfaces\nby Eyal Lubetzky (
Courant) as part of Oxford discrete mathematics and probability seminar\n\
n\nAbstract\nDobrushin (1972) showed that\, at low enough temperatures\, t
he interface of the 3D Ising model - the random surface separating the plu
s and minus phases above and below the xy-plane - is localized: it has O(1
) height fluctuations above a fixed point\, and its maximum height Mn on a
box of side length n is OP(log n). We study this interface and derive a s
hape theorem for its ``pillars'' conditionally on reaching an atypically l
arge height. We use this to analyze the maximum height Mn of the interface
\, and prove that at low temperature Mn/log n converges to cβ in probabil
ity. Furthermore\, the sequence (Mn - E[Mn])n≥1 is tight\, and even thou
gh this sequence does not converge\, its subsequential limits satisfy unif
orm Gumbel tails bounds.\nJoint work with Reza Gheissari.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Catherine Greenhill (UNSW)
DTSTART:20200526T083000Z
DTEND:20200526T093000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/13
DESCRIPTION:Title: The small subgraph conditioning method and hypergraphs\nb
y Catherine Greenhill (UNSW) as part of Oxford discrete mathematics and pr
obability seminar\n\n\nAbstract\nThe small subgraph conditioning method is
an analysis of variance technique which was introduced by Robinson and Wo
rmald in 1992\, in their proof that almost all cubic graphs are Hamiltonia
n. The method has been used to prove many structural results about random
regular graphs\, mostly to show that a certain substructure is present wit
h high probability. I will discuss some applications of the small subgraph
conditioning method to hypergraphs\, and describe a subtle issue which is
absent in the graph setting.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Wood (Monash)
DTSTART:20200526T100000Z
DTEND:20200526T110000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/14
DESCRIPTION:Title: Subgraph densities in a surface\nby David Wood (Monash) a
s part of Oxford discrete mathematics and probability seminar\n\n\nAbstrac
t\nWe study the following question at the intersection of extremal and str
uctural graph theory. Given a fixed graph H that embeds in a fixed surface
Σ\, what is the maximum number of copies of H in an n-vertex graph that
embeds in Σ? Exact answers to this question are known for specific graphs
H when Σ is the sphere. We aim for more general\, albeit less precise\,
results. We show that the answer to the above question is Θ(nf(H))\, wher
e f(H) is a graph invariant called the `flap-number' of H\, which is indep
endent of Σ. This simultaneously answers two open problems posed by Eppst
ein (1993). When H is a complete graph we give more precise answers. This
is joint work with Tony Huynh and Gwenaël Joret\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dana Randall (Georgia Tech)
DTSTART:20200609T130000Z
DTEND:20200609T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/15
DESCRIPTION:Title: Markov Chains for Programmable Active Matter\nby Dana Ran
dall (Georgia Tech) as part of Oxford discrete mathematics and probability
seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Will Perkins (UIC)
DTSTART:20200609T140000Z
DTEND:20200609T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/16
DESCRIPTION:Title: First-order phase transitions and efficient sampling algorith
ms\nby Will Perkins (UIC) as part of Oxford discrete mathematics and p
robability seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Allan Sly (Princeton)
DTSTART:20200609T153000Z
DTEND:20200609T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/17
DESCRIPTION:Title: Replica Symmetry Breaking for Random Regular NAESAT\nby A
llan Sly (Princeton) as part of Oxford discrete mathematics and probabilit
y seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wojciech Samotij (Tel Aviv)
DTSTART:20200602T130000Z
DTEND:20200602T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/18
DESCRIPTION:Title: An entropy proof of the Erdős-Kleitman-Rothschild theorem.\nby Wojciech Samotij (Tel Aviv) as part of Oxford discrete mathematics
and probability seminar\n\n\nAbstract\nWe say that a graph G is H-free if
G does not contain H as a (not necessarily induced) subgraph. For a positi
ve integer n\, denote by ex(n\,H) the largest number of edges in an H-free
graph with n vertices (the Turán number of H). The classical theorem of
Erdős\, Kleitman\, and Rothschild states that\, for every r≥3\, there a
re 2ex(n\,H)+o(n2) many Kr-free graphs with vertex set {1\,…\, n}. There
exist (at least) three different derivations of this estimate in the lite
rature: an inductive argument based on the Kővári-Sós-Turán theorem (a
nd its generalisation to hypergraphs due to Erdős)\, a proof based on Sze
merédi's regularity lemma\, and an argument based on the hypergraph conta
iner theorems. In this talk\, we present yet another proof of this bound t
hat exploits connections between entropy and independence. This argument i
s an adaptation of a method developed in a joint work with Gady Kozma\, To
m Meyerovitch\, and Ron Peled that studied random metric spaces.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean Bertoin (University of Zürich)
DTSTART:20200602T143000Z
DTEND:20200602T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/19
DESCRIPTION:Title: Scaling exponents of step-reinforced random walks\nby Jea
n Bertoin (University of Zürich) as part of Oxford discrete mathematics a
nd probability seminar\n\n\nAbstract\nLet X1\, … be i.i.d. copies of som
e real random variable X. For any ε2\, ε3\, … in {0\,1}\, a basic algo
rithm introduced by H.A. Simon yields a reinforced sequence X̂1\, X̂2\,
… as follows. If εn=0\, then X̂n is a uniform random sample from X̂1\
, …\, X̂n-1\; otherwise X̂n is a new independent copy of X. The purpos
e of this talk is to compare the scaling exponent of the usual random walk
S(n)=X1 +… + Xn with that of its step reinforced version Ŝ(n)=X̂1+
… + X̂n. Depending on the tail of X and on asy\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rob Morris (IMPA)
DTSTART:20200331T130000Z
DTEND:20200331T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/20
DESCRIPTION:Title: Erdős covering systems\nby Rob Morris (IMPA) as part of
Oxford discrete mathematics and probability seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Louigi Addario-Berry (McGill)
DTSTART:20200407T130000Z
DTEND:20200407T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/21
DESCRIPTION:Title: Hipster random walks and their ilk\nby Louigi Addario-Ber
ry (McGill) as part of Oxford discrete mathematics and probability seminar
\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Janos Pach (Rényi Institute)
DTSTART:20201006T130000Z
DTEND:20201006T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/22
DESCRIPTION:Title: The Schur-Erdős problem for graphs of bounded dimension\
nby Janos Pach (Rényi Institute) as part of Oxford discrete mathematics a
nd probability seminar\n\n\nAbstract\nThere is a growing body of results i
n extremal combinatorics and Ramsey theory which give better bounds or str
onger conclusions under the additional assumption of bounded VC-dimension.
Schur and Erdős conjectured that there exists a suitable constant $c$ wi
th the property that every graph with at least $2^{cm}$ vertices\, whose e
dges are colored by $m$ colors\, contains a monochromatic triangle. We pro
ve this conjecture for edge-colored graphs such that the set system induce
d by the neighborhoods of the vertices with respect to each color class ha
s bounded VC-dimension. This result is best possible up to the value of $c
$.\n Joint work with Jacob Fox and Andrew Suk.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nina Holden (ETH)
DTSTART:20201006T143000Z
DTEND:20201006T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/23
DESCRIPTION:Title: Liouville quantum gravity with matter central in (1\,25): a p
robabilistic approach\nby Nina Holden (ETH) as part of Oxford discrete
mathematics and probability seminar\n\n\nAbstract\nLiouville quantum grav
ity (LQG) is a theory of random fractal surfaces with origin in the physic
s literature in the 1980s. Most literature is about LQG with matter centra
l charge $c\\in(-\\infty\,1]$. We study a discretization of LQG which make
s sense for all c\\in(-\\infty\,25)$. Based on a joint work with Gwynne\,
Pfeffer\, and Remy.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asaf Nachmias (Tel Aviv)
DTSTART:20201013T130000Z
DTEND:20201013T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/24
DESCRIPTION:Title: The local limit of uniform spanning trees\nby Asaf Nachmi
as (Tel Aviv) as part of Oxford discrete mathematics and probability semin
ar\n\n\nAbstract\nLet $G_n$ be a sequence of finite\, simple\, connected\,
regular graphs with degrees tending to infinity and let Tn be a uniformly
drawn spanning tree of $G_n$. In joint work with Yuval Peres we show that
the local limit of $T_n$ is the Poisson(1) branching process conditioned
to survive forever (that is\, the asymptotic frequency of the appearance o
f any small subtree is given by the branching process). The proof is based
on electric network theory and I hope to show most of it.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Caroline Terry (Ohio State)
DTSTART:20201013T143000Z
DTEND:20201013T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/25
DESCRIPTION:Title: Speeds of hereditary properties and mutual algebricity\nb
y Caroline Terry (Ohio State) as part of Oxford discrete mathematics and p
robability seminar\n\n\nAbstract\nA hereditary graph property is a class o
f finite graphs closed under isomorphism and induced subgraphs. Given a he
reditary graph property $H$\, the speed of $H$ is the function which sends
an integer n to the number of distinct elements in $H$ with underlying se
t $\\{1\,...\,n\\}$. Not just any function can occur as the speed of hered
itary graph property. Specifically\, there are discrete "jumps" in the pos
sible speeds. Study of these jumps began with work of Scheinerman and Zito
in the 90's\, and culminated in a series of papers from the 2000's by Bal
ogh\, Bollobás\, and Weinreich\, in which essentially all possible speeds
of a hereditary graph property were characterized. In contrast to this\,
many aspects of this problem in the hypergraph setting remained unknown. I
n this talk we present new hypergraph analogues of many of the jumps from
the graph setting\, specifically those involving the polynomial\, exponent
ial\, and factorial speeds. The jumps in the factorial range turned out to
have surprising connections to the model theoretic notion of mutual algeb
ricity\, which we also discuss. This is joint work with Chris Laskowski.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Croydon (Kyoto)
DTSTART:20201020T080000Z
DTEND:20201020T090000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/26
DESCRIPTION:Title: Scaling limits of the two- and three-dimensional uniform span
ning trees\nby David Croydon (Kyoto) as part of Oxford discrete mathem
atics and probability seminar\n\n\nAbstract\nI will introduce recent work
on the two- and three-dimensional uniform spanning trees (USTs) that estab
lish the laws of these random objects converge under rescaling in a space
whose elements are measured\, rooted real trees\, continuously embedded in
to Euclidean space. (In the three-dimensional case\, the scaling result is
currently only known along a particular scaling sequence.) I will also di
scuss various properties of the intrinsic metrics and measures of the limi
ting spaces\, including their Hausdorff dimension\, as well as the scaling
limits of the random walks on the two- and three-dimensional USTs. In the
talk\, I will attempt to emphasise where the differences lie between the
two cases\, and in particular the additional challenges that arise when it
comes to the three-dimensional model.\n The two-dimensional results ar
e joint with Martin Barlow (UBC) and Takashi Kumagai (Kyoto). The three-di
mensional results are joint with Omer Angel (UBC) and Sarai Hernandez-Torr
es (UBC).\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anita Liebenau (UNSW)
DTSTART:20201020T093000Z
DTEND:20201020T103000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/27
DESCRIPTION:Title: The threshold bias of the clique-factor game\nby Anita Li
ebenau (UNSW) as part of Oxford discrete mathematics and probability semin
ar\n\n\nAbstract\nLet $r>3$ be an integer and consider the following game
on the complete graph $K_n$ for $n$ a multiple of $r$: Two players\, Maker
and Breaker\, alternately claim previously unclaimed edges of $K_n$ such
that in each turn Maker claims one and Breaker claims $b$ edges. Maker win
s if her graph contains a $K_r$-factor\, that is a collection of $n/r$ ver
tex-disjoint copies of $K_r$\, and Breaker wins otherwise. In other words\
, we consider the $b$-biased $K_r$-factor Maker-Breaker game. We show that
the threshold bias for this game is of order $n^2/(r+2)$. This makes a st
ep towards determining the threshold bias for making bounded-degree spanni
ng graphs and extends a result of Allen\, Böttcher\, Kohayakawa\, Naves a
nd Person who resolved the case $r=3$ or $4$ up to a logarithmic factor.\n
Joint work with Rajko Nenadov.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Igor Kortchemski (Ecole Polytechnique)
DTSTART:20201027T140000Z
DTEND:20201027T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/28
DESCRIPTION:Title: The geometry of random minimal factorizations of a long cycle
\nby Igor Kortchemski (Ecole Polytechnique) as part of Oxford discrete
mathematics and probability seminar\n\n\nAbstract\nWe will be interested
in the structure of random typical minimal factorizations of the n-cycle i
nto transpositions\, which are factorizations of $(1\,\\ldots\,n)$ as a pr
oduct of $n-1$ transpositions. We shall establish a phase transition when
a certain amount of transpositions have been read one after the other. One
of the main tools is a limit theorem for two-type Bienaymé-Galton-Watson
trees conditioned on having given numbers of vertices of both types\, whi
ch is of independent interest. This is joint work with Valentin Féray.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luke Postle (Waterloo)
DTSTART:20201027T153000Z
DTEND:20201027T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/29
DESCRIPTION:Title: Further progress towards Hadwiger's conjecture\nby Luke P
ostle (Waterloo) as part of Oxford discrete mathematics and probability se
minar\n\n\nAbstract\nIn 1943\, Hadwiger conjectured that every graph with
no Kt minor is $(t-1)$-colorable for every $t\\geq 1$. In the 1980s\, Kost
ochka and Thomason independently proved that every graph with no $K_t$ min
or has average degree $O(t(\\log t)^{1/2})$ and hence is $O(t(\\log t)^{1/
2)}$-colorable. Recently\, Norin\, Song and I showed that every graph with
no $K_t$ minor is $O(t(\\log t)^\\beta)$-colorable for every $\\beta > 1/
4$\, making the first improvement on the order of magnitude of the $O(t(\\
log t)^{1/2})$ bound. Here we show that every graph with no $K_t$ minor is
$O(t (\\log t)^\\beta)$-colorable for every $\\beta > 0$\; more specifica
lly\, they are $O(t (\\log \\log t)^6)$-colorable.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julian Sahasrabudhe (Cambridge)
DTSTART:20201103T140000Z
DTEND:20201103T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/30
DESCRIPTION:Title: Combinatorics from the zeros of polynomials\nby Julian Sa
hasrabudhe (Cambridge) as part of Oxford discrete mathematics and probabil
ity seminar\n\n\nAbstract\nLet $X$ be a random variable\, taking values in
$\\{1\,…\,n\\}$\, with standard deviation $\\sigma$ and let $f_X$ be it
s probability generating function. Pemantle conjectured that if $\\sigma$
is large and $f_X$ has no roots close to 1 in the complex plane then $X$ m
ust approximate a normal distribution. In this talk\, I will discuss a com
plete resolution of Pemantle's conjecture. As an application\, we resolve
a conjecture of Ghosh\, Liggett and Pemantle by proving a multivariate cen
tral limit theorem for\, so called\, strong Rayleigh distributions. I will
also discuss how these sorts of results shed light on random variables th
at arise naturally in combinatorial settings. This talk is based on joint
work with Marcus Michelen.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shoham Letzter (UCL)
DTSTART:20201103T153000Z
DTEND:20201103T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/31
DESCRIPTION:Title: An improvement on Łuczak's connected matchings method\nb
y Shoham Letzter (UCL) as part of Oxford discrete mathematics and probabil
ity seminar\n\n\nAbstract\nA connected matching is a matching contained in
a connected component. A well-known method due to Łuczak reduces problem
s about monochromatic paths and cycles in complete graphs to problems abou
t monochromatic matchings in almost complete graphs. We show that these ca
n be further reduced to problems about monochromatic connected matchings i
n complete graphs.\n \nI will describe Łuczak's reduction\, introduce
the new reduction\, and mention potential applications of the improved met
hod.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent Beffara (Grenoble)
DTSTART:20201110T140000Z
DTEND:20201110T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/32
DESCRIPTION:Title: Critical behavior without FKG\nby Vincent Beffara (Grenob
le) as part of Oxford discrete mathematics and probability seminar\n\n\nAb
stract\nI will present work in progress with D. Gayet and F. Pouran (Greno
ble) to establish Russo-Seymour-Welsh (RSW) estimates for 2d statistical m
echanics models that do not satisfy the FKG inequality. RSW states that cr
itical percolation has no characteristic length\, in the sense that large
rectangles are crossed by an open path with a probability that is bounded
below by a function of their shape\, but uniformly in their size\; this en
sures the polynomial decay of many relevant quantities and opens the way t
o deeper understanding of the critical features of the model. All the stan
dard proofs of RSW rely on the FKG inequality\, i.e. on the positive corre
lation between increasing events\; we establish the stability of RSW under
small perturbations that do not preserve FKG\, which extends it for insta
nce to the high-temperature anti-ferromagnetic Ising model.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Hutchcroft (Cambridge)
DTSTART:20201110T153000Z
DTEND:20201110T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/33
DESCRIPTION:Title: Power-law bounds for critical long-range percolation\nby
Tom Hutchcroft (Cambridge) as part of Oxford discrete mathematics and prob
ability seminar\n\n\nAbstract\nIn long-range percolation on $\\mathbb{Z}^d
$\, each potential edge $\\{x\,y\\}$ is included independently at random w
ith probability roughly $\\beta\\|x-y\\|-d-\\alpha$\, where $\\alpha > 0
$ controls how long-range the model is and $\\beta > 0$ is an intensity p
arameter. The smaller $\\alpha$ is\, the easier it is for very long edges
to appear. We are normally interested in fixing $\\alpha$ and studying t
he phase transition that occurs as $\\beta$ is increased and an infinite
cluster emerges. Perhaps surprisingly\, the phase transition for long-rang
e percolation is much better understood than that of nearest neighbour per
colation\, at least when $\\alpha$ is small: It is a theorem of Noam Berg
er that if $\\alpha < d$ then the phase transition is continuous\, meanin
g that there are no infinite clusters at the critical value of $\\beta$. (
Proving the analogous result for nearest neighbour percolation is a notori
ous open problem!) In my talk I will describe a new\, quantitative proof o
f Berger's theorem that yields power-law upper bounds on the distribution
of the cluster of the origin at criticality.\n As a part of this proo
f\, I will describe a new universal inequality stating that on any graph\,
the maximum size of a percolation cluster is of the same order as its med
ian with high probability. This inequality can also be used to give stream
lined new proofs of various classical results on e.g. Erdős-Rényi random
graphs\, which I will hopefully have time to talk a little bit about also
.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yuval Peled (Courant)
DTSTART:20201117T140000Z
DTEND:20201117T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/34
DESCRIPTION:Title: Minimum weight disk triangulations and fillings\nby Yuval
Peled (Courant) as part of Oxford discrete mathematics and probability se
minar\n\n\nAbstract\nWe study the minimum total weight of a disk triangula
tion using any number of vertices out of $\\{1\,..\,n\\}$ where the bounda
ry is fixed and the $n \\choose 3$ triangles have independent rate-1 expon
ential weights. We show that\, with high probability\, the minimum weight
is equal to $(c+o(1))n-1/2\\log n$ for an explicit constant $c$. Further\,
we prove that\, with high probability\, the minimum weights of a homologi
cal filling and a homotopical filling of the cycle $(123)$ are both attain
ed by the minimum weight disk triangulation. We will discuss a related ope
n problem concerning simple-connectivity of random simplicial complexes\,
where a similar phenomenon is conjectured to hold. Based on joint works wi
th Itai Benjamini\, Eyal Lubetzky\, and Zur Luria.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ron Rosenthal (Technion)
DTSTART:20201117T153000Z
DTEND:20201117T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/35
DESCRIPTION:Title: Random Steiner complexes and simplical spanning trees\nby
Ron Rosenthal (Technion) as part of Oxford discrete mathematics and proba
bility seminar\n\n\nAbstract\nA spanning tree of $G$ is a subgraph of $G$
with the same vertex set as $G$ that is a tree. In 1981\, McKay proved an
asymptotic result regarding the number of spanning trees in random $k$-reg
ular graphs\, showing that the number of spanning trees $\\kappa_1(G_n)$ i
n a random $k$-regular graph on $n$ vertices satisfies $\\lim_{n \\to \\in
fty} (\\kappa_1(G_n))^{1/n} = c_{1\,k}$ in probability\, where $c_{1\,k} =
(k-1)^{k-1} (k^2-2k)^{-(k-2)/2}$. \n\nIn this talk we will discuss a high
-dimensional of the matching model for simplicial complexes\, known as ran
dom Steiner complexes. In particular\, we will prove a high-dimensional co
unterpart of McKay's result and discuss the local limit of such random com
plexes. \nBased on a joint work with Lior Tenenbaum.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Holroyd (Bristol)
DTSTART:20201124T140000Z
DTEND:20201124T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/36
DESCRIPTION:Title: Matching Random Points\nby Alexander Holroyd (Bristol) as
part of Oxford discrete mathematics and probability seminar\n\n\nAbstract
\nWhat is fairness\, and to what extent is it practically achievable? I'll
talk about a simple mathematical model under which one might hope to unde
rstand such questions. Red and blue points occur as independent homogeneou
s Poisson processes of equal intensity in Euclidean space\, and we try to
match them to each other. We would like to minimize the sum of a some func
tion (say\, a power\, $\\gamma$) of the distances between matched pairs. T
his does not make sense\, because the sum is infinite\, so instead we sati
sfy ourselves with minimizing *locally*. If the points are interpreted as
agents who would like to be matched as close as possible\, the parameter $
\\gamma$ encodes a measure of fairness - large $\\gamma$ means that we try
to avoid occasional very bad outcomes (long edges)\, even if that means i
nconvenience to others - small $\\gamma$ means everyone is in it for thems
elves.\n In dimension 1 we have a reasonably complete picture\, with a
phase transition at $\\gamma=1$. For $\\gamma<1$ there is a unique minimal
matching\, while for $\\gamma>1$ there are multiple matchings but no stat
ionary solution. In higher dimensions\, even existence is not clear in all
cases.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gwenaël Joret (Université Libre de Bruxelles)
DTSTART:20201124T153000Z
DTEND:20201124T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/37
DESCRIPTION:Title: Sparse universal graphs for planarity\nby Gwenaël Joret
(Université Libre de Bruxelles) as part of Oxford discrete mathematics an
d probability seminar\n\n\nAbstract\nThis talk will focus on the following
two related problems:\n (1) What is the minimum number of edges in a
graph containing all $n$-vertex planar graphs as subgraphs? A simple con
struction of Babai\, Erdos\, Chung\, Graham\, and Spencer (1982) has $O(n^
{3/2})$ edges\, which is the best known upper bound.\n (2) What is th
e minimum number of *vertices* in a graph containing all $n$-vertex planar
graphs as *induced* subgraphs? Here steady progress has been achieved ove
r the years\, culminating in a $O(n^{4/3})$ bound due to Bonamy\, Gavoille
\, and Pilipczuk (2019).\n As it turns out\, a bound of $n^{1+o(1)}$
can be achieved for each of these two problems. The two constructions are
somewhat different but are based on a common technique. In this talk I wil
l first give a gentle introduction to the area and then sketch these const
ructions. The talk is based on joint works with Vida Dujmović\, Louis Esp
eret\, Cyril Gavoille\, Piotr Micek\, and Pat Morin.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Emmanuel Breuillard (Cambridge)
DTSTART:20210119T143000Z
DTEND:20210119T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/38
DESCRIPTION:Title: A subspace theorem for manifolds\nby Emmanuel Breuillard
(Cambridge) as part of Oxford discrete mathematics and probability seminar
\n\n\nAbstract\nThe Schmidt subspace theorem is a far-reaching generalizat
ion of the Thue-Siegel-Roth theorem in diophantine approximation. In this
talk I will give an interpretation of Schmidt's subspace theorem in terms
of the dynamics of diagonal flows on homogeneous spaces and describe how t
he exceptional subspaces arise from certain rational Schubert varieties as
sociated to the family of linear forms through the notion of Harder-Narasi
mhan filtration and an associated slope formalism. This geometric understa
nding opens the way to a natural generalization of Schmidt's theorem to th
e setting of diophantine approximation on submanifolds of $GL_d$\, which i
s our main result. In turn this allows us to recover and generalize the ma
in results of Kleinbock and Margulis regarding diophantine exponents of su
bmanifolds. I will also mention an application to diophantine approximatio
n on Lie groups. Joint work with Nicolas de Saxcé.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Artem Chernikov (UCLA)
DTSTART:20210119T160000Z
DTEND:20210119T170000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/39
DESCRIPTION:Title: Hypergraph regularity and higher arity VC-dimension\nby A
rtem Chernikov (UCLA) as part of Oxford discrete mathematics and probabili
ty seminar\n\n\nAbstract\nWe generalize the fact that all graphs omitting
a fixed finite bipartite graph can be uniformly approximated by rectangles
(Alon-Fischer-Newman\, Lovász-Szegedy)\, showing that hypergraphs omitti
ng a fixed finite $(k+1)$-partite $(k+1)$-uniform hypergraph can be approx
imated by $k$-ary cylinder sets. In particular\, in the decomposition give
n by hypergraph regularity one only needs the first $k$ levels: such hyper
graphs can be approximated using sets of vertices\, sets of pairs\, and so
on up to sets of $k$-tuples\, and on most of the resulting $k$-ary cylind
er sets\, the density is either close to 0 or close to 1. Moreover\, exist
ence of such approximations uniformly under all measures on the vertices i
s a characterization. Our proof uses a combination of analytic\, combinato
rial and model-theoretic methods\, and involves a certain higher arity gen
eralization of the epsilon-net theorem from VC-theory. Joint work with He
nry Towsner.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Richard Montgomery (Birmingham)
DTSTART:20210126T140000Z
DTEND:20210126T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/40
DESCRIPTION:Title: A solution to Erdős and Hajnal's odd cycle problem\nby R
ichard Montgomery (Birmingham) as part of Oxford discrete mathematics and
probability seminar\n\n\nAbstract\nI will discuss how to construct cycles
of many different lengths in graphs\, in particular answering the followin
g two problems on odd and even cycles. Erdős and Hajnal asked in 1981 whe
ther the sum of the reciprocals of the odd cycle lengths in a graph diverg
es as the chromatic number increases\, while\, in 1984\, Erdős asked whet
her there is a constant $C$ such that every graph with average degree at l
east $C$ contains a cycle whose length is a power of 2.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Noga Alon (Princeton)
DTSTART:20210126T153000Z
DTEND:20210126T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/41
DESCRIPTION:Title: Random friends walking on random graphs\nby Noga Alon (Pr
inceton) as part of Oxford discrete mathematics and probability seminar\n\
n\nAbstract\nLet $X$ and $Y$ be two $n$-vertex graphs. Identify the vertic
es of $Y$ with $n$ people\, any two of whom are either friends or stranger
s (according to the edges and non-edges in $Y$)\, and imagine that these p
eople are standing one at each vertex of $X$. At each point in time\, two
friends standing at adjacent vertices of $X$ may swap places\, but two str
angers may not. The friends-and-strangers graph $FS(X\,Y)$ has as its vert
ex set the collection of all configurations of people standing on the vert
ices of $X$\, where two configurations are adjacent when they are related
via a single friendly swap. This provides a common generalization for the
famous 15-puzzle\, transposition Cayley graphs of symmetric groups\, and e
arly work of Wilson and of Stanley.\nI will describe several recent result
s and open problems addressing the extremal and typical aspects of the not
ion\, focusing on the result that the threshold probability for connectedn
ess of $FS(X\,Y)$ for two independent binomial random graphs $X$ and $Y$ i
n $G(n\,p)$ is $p=p(n)=n-1/2+o(1)$.\nJoint work with Colin Defant and Noah
Kravitz.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lisa Sauermann (IAS)
DTSTART:20210202T140000Z
DTEND:20210202T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/42
DESCRIPTION:Title: On the extension complexity of low-dimensional polytopes\
nby Lisa Sauermann (IAS) as part of Oxford discrete mathematics and probab
ility seminar\n\n\nAbstract\nIt is sometimes possible to represent a compl
icated polytope as a projection of a much simpler polytope. To quantify th
is phenomenon\, the extension complexity of a polytope $P$ is defined to b
e the minimum number of facets in a (possibly higher-dimensional) polytope
from which $P$ can be obtained as a (linear) projection. In this talk\, w
e discuss some results on the extension complexity of random $d$-dimension
al polytopes (obtained as convex hulls of random points on either on the u
nit sphere or in the unit ball)\, and on the extension complexity of polyg
ons with all vertices on a common circle. Joint work with Matthew Kwan and
Yufei Zhao\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Stephenson (Sheffield)
DTSTART:20210209T140000Z
DTEND:20210209T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/43
DESCRIPTION:Title: The scaling limit of a critical random directed graph\nby
Robin Stephenson (Sheffield) as part of Oxford discrete mathematics and p
robability seminar\n\n\nAbstract\nWe consider the random directed graph $D
(n\,p)$ with vertex set $\\{1\,2\,…\,n\\}$ in which each of the $n(n-1)$
possible directed edges is present independently with probability $p$. We
are interested in the strongly connected components of this directed grap
h. A phase transition for the emergence of a giant strongly connected comp
onent is known to occur at $p = 1/n$\, with critical window $p = 1/n + \\l
ambda n-4/3$ for $\\lambda \\in \\mathbb{R}$. We show that\, within this c
ritical window\, the strongly connected components of $D(n\,p)$\, ranked i
n decreasing order of size and rescaled by $n-1/3$\, converge in distribut
ion to a sequence $(C_1\,C_2\,\\ldots)$ of finite strongly connected direc
ted multigraphs with edge lengths which are either 3-regular or loops. The
convergence occurs in the sense of an $L^1$ sequence metric for which two
directed multigraphs are close if there are compatible isomorphisms betwe
en their vertex and edge sets which roughly preserve the edge lengths. Our
proofs rely on a depth-first exploration of the graph which enables us to
relate the strongly connected components to a particular spanning forest
of the undirected Erdős-Rényi random graph $G(n\,p)$\, whose scaling lim
it is well understood. We show that the limiting sequence $(C_1\,C_2\,\\ld
ots)$ contains only finitely many components which are not loops. If we ig
nore the edge lengths\, any fixed finite sequence of 3-regular strongly co
nnected directed multigraphs occurs with positive probability.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nati Linial (Hebrew University of Jerusalem)
DTSTART:20210216T140000Z
DTEND:20210216T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/44
DESCRIPTION:Title: Geodesic Geometry on Graphs\nby Nati Linial (Hebrew Unive
rsity of Jerusalem) as part of Oxford discrete mathematics and probability
seminar\n\n\nAbstract\nWe investigate a graph theoretic analog of geodesi
c geometry. In a graph $G=(V\,E)$ we consider a system of paths $P=\\{P_{u
\,v}| u\,v\\in V\\}$ where $P_{u\,v}$ connects vertices $u$ and $v$. This
system is consistent in that if vertices $y\,z$ are in $P_{u\,v}$\, then t
he sub-path of $P_{u\,v}$ between them coincides with $P_{y\,z}$. A map $w
:E\\to(0\,\\infty)$ is said to induce $P$ if for every $u\,v\\in V$ the pa
th $P_{u\,v}$ is $w$-geodesic. We say that $G$ is metrizable if every cons
istent path system is induced by some such $w$. As we show\, metrizable gr
aphs are very rare\, whereas there exist infinitely many 2-connected metri
zable graphs.\nThis is the MSc thesis of Daniel Cizma done under my guidan
ce.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Justin Salez (Université Paris-Dauphine)
DTSTART:20210302T140000Z
DTEND:20210302T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/45
DESCRIPTION:Title: Sparse expanders have negative Ollivier-Ricci curvature\n
by Justin Salez (Université Paris-Dauphine) as part of Oxford discrete ma
thematics and probability seminar\n\n\nAbstract\nWe prove that bounded-deg
ree expanders with non-negative Ollivier-Ricci curvature do not exist\, th
ereby solving a long-standing open problem suggested by Naor and Milman an
d publicized by Ollivier (2010). In fact\, this remains true even if we al
low for a vanishing proportion of large degrees\, large eigenvalues\, and
negatively-curved edges. To establish this\, we work directly at the level
of Benjamini-Schramm limits. More precisely\, we exploit the entropic cha
racterization of the Liouville property on stationary random graphs to sho
w that non-negative curvature and spectral expansion are incompatible 'at
infinity'. We then transfer this result to finite graphs via local weak co
nvergence and a relative compactness argument. We believe that this 'local
weak limit' approach to mixing properties of Markov chains will have many
other applications.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Perla Sousi (Cambridge)
DTSTART:20210302T153000Z
DTEND:20210302T170000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/46
DESCRIPTION:Title: The uniform spanning tree in 4 dimensions\nby Perla Sousi
(Cambridge) as part of Oxford discrete mathematics and probability semina
r\n\n\nAbstract\nA uniform spanning tree of $\\mathbb{Z}^4$ can be thought
of as the "uniform measure" on trees of $\\mathbb{Z}^4$. The past of 0 in
the uniform spanning tree is the finite component that is disconnected fr
om infinity when 0 is deleted from the tree. We establish the logarithmic
corrections to the probabilities that the past contains a path of length $
n$\, that it has volume at least $n$ and that it reaches the boundary of t
he box of side length $n$ around 0. Dimension 4 is the upper critical dime
nsion for this model in the sense that in higher dimensions it exhibits "m
ean-field" critical behaviour. An important part of our proof is the study
of the Newtonian capacity of a loop erased random walk in 4 dimensions. T
his is joint work with Tom Hutchcroft.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bénédicte Haas (Paris 13)
DTSTART:20210309T140000Z
DTEND:20210309T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/47
DESCRIPTION:Title: Tail asymptotics for extinction times of self-similar fragmen
tations\nby Bénédicte Haas (Paris 13) as part of Oxford discrete mat
hematics and probability seminar\n\n\nAbstract\nSelf-similar fragmentation
processes are random models for particles that are subject to successive
fragmentations. When the index of self-similarity is negative the fragment
ations intensify as the masses of particles decrease. This leads to a shat
tering phenomenon\, where the initial particle is entirely reduced to dust
- a set of zero-mass particles - in finite time which is what we call the
extinction time. Equivalently\, these extinction times may be seen as hei
ghts of continuous compact rooted trees or scaling limits of heights of se
quences of discrete trees. Our objective is to set up precise bounds for t
he large time asymptotics of the tail distributions of these extinction ti
mes.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nathanaël Berestycki (Vienna)
DTSTART:20210202T153000Z
DTEND:20210202T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/48
DESCRIPTION:Title: Free boundary dimers: random walk representation and scaling
limit\nby Nathanaël Berestycki (Vienna) as part of Oxford discrete ma
thematics and probability seminar\n\n\nAbstract\nThe dimer model\, a class
ical model of statistical mechanics\, is the uniform distribution on perfe
ct matchings of a graph. In two dimensions\, one can define an associated
height function which turns the model into a random surface (with specifie
d boundary conditions). In the 1960s\, Kasteleyn and Temperley/Fisher foun
d an exact "solution" to the model\, computing the correlations in terms o
f a matrix called the Kasteleyn matrix. This exact solvability was the sta
rting point for the breakthrough work of Kenyon (2000) who proved that the
centred height function converges to the Dirichlet (or zero boundary cond
itions) Gaussian free field. This was the first proof of conformal invaria
nce in statistical mechanics.\n\nIn this talk\, I will focus on a natural
modification of the model where one allows the vertices on the boundary of
the graph to remain unmatched: this is the so-called monomer-dimer model\
, or dimer model with free boundary conditions. The main result that we ob
tain is that the scaling limit of the height function of the monomer-dimer
model in the upper half-plane is the Neumann (or free boundary conditions
) Gaussian free field. Key to this result is a somewhat miraculous random
walk representation for the inverse Kasteleyn matrix\, which I hope to dis
cuss.\n\nJoint work with Marcin Lis (Vienna) and Wei Qian (Paris).\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ramon van Handel (Princeton)
DTSTART:20210216T153000Z
DTEND:20210216T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/49
DESCRIPTION:Title: Some unusual extremal problems in convexity and combinatorics
\nby Ramon van Handel (Princeton) as part of Oxford discrete mathemati
cs and probability seminar\n\n\nAbstract\nIt is a basic fact of convexity
that the volume of convex bodies is a polynomial\, whose coefficients cont
ain many familiar geometric parameters as special cases. A fundamental res
ult of convex geometry\, the Alexandrov-Fenchel inequality\, states that t
hese coefficients are log-concave. This proves to have striking connection
s with other areas of mathematics: for example\, the appearance of log-con
cave sequences in many combinatorial problems may be understood as a conse
quence of the Alexandrov-Fenchel inequality and its algebraic analogues.\n
\nThere is a long-standing problem surrounding the Alexandrov-Fenchel ineq
uality that has remained open since the original works of Minkowski (1903)
and Alexandrov (1937): in what cases is equality attained? In convexity\,
this question corresponds to the solution of certain unusual isoperimetri
c problems\, whose extremal bodies turn out to be numerous and strikingly
bizarre. In combinatorics\, an answer to this question would provide nontr
ivial information on the type of log-concave sequences that can arise in c
ombinatorial applications. In recent work with Y. Shenfeld\, we succeeded
to settle the equality cases completely in the setting of convex polytopes
. I will aim to describe this result\, and to illustrate its potential com
binatorial implications through a question of Stanley on the combinatorics
of partially ordered sets.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vida Dujmović (Ottawa)
DTSTART:20210209T153000Z
DTEND:20210209T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/50
DESCRIPTION:Title: Product structure theory and its applications\nby Vida Du
jmović (Ottawa) as part of Oxford discrete mathematics and probability se
minar\n\n\nAbstract\nI will introduce product structure theory of graphs a
nd show how families of graphs that have such a structure admit short adja
cency labeling scheme and small induced universal graphs. Time permitting\
, I will talk about another recent application of product structure theory
\, namely vertex ranking (coloring).\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Corrine Yap (Rutgers)
DTSTART:20210309T153000Z
DTEND:20210309T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/51
DESCRIPTION:Title: A Topological Turán Problem\nby Corrine Yap (Rutgers) as
part of Oxford discrete mathematics and probability seminar\n\n\nAbstract
\nThe classical Turán problem asks: given a graph $H$\, how many edges ca
n an $4n$-vertex graph have while containing no isomorphic copy of $H$? By
viewing $(k+1)$-uniform hypergraphs as $k$-dimensional simplicial complex
es\, we can ask a topological version (first posed by Nati Linial): given
a $k$-dimensional simplicial complex $S$\, how many facets can an $n$-vert
ex $k$-dimensional simplicial complex have while containing no homeomorphi
c copy of $S$? Until recently\, little was known for $k > 2$. In this talk
\, we give an answer for general $k$\, by way of dependent random choice a
nd the combinatorial notion of a trace-bounded hypergraph. Joint work with
Jason Long and Bhargav Narayanan.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Guillem Perarnau (Universitat Politècnica de Catalunya)
DTSTART:20210427T130000Z
DTEND:20210427T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/52
DESCRIPTION:Title: Maximum stationary values in directed random graphs\nby G
uillem Perarnau (Universitat Politècnica de Catalunya) as part of Oxford
discrete mathematics and probability seminar\n\n\nAbstract\nIn this talk w
e will consider the extremal values of the stationary distribution of the
sparse directed configuration model. Under the assumption of linear $(2+\\
eta)$-moments on the in-degrees and of bounded out-degrees\, we obtain tig
ht comparisons between the maximum value of the stationary distribution an
d the maximum in-degree. Under the further assumption that the order stati
stics of the in-degrees have power-law behavior\, we show that the upper t
ail of the stationary distribution also has power-law behavior with the sa
me index. Moreover\, these results extend to the PageRank scores of the mo
del\, thus confirming a version of the so-called power-law hypothesis. Joi
nt work with Xing Shi Cai\, Pietro Caputo and Matteo Quattropani.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roberto Oliveira (IMPA)
DTSTART:20210427T143000Z
DTEND:20210427T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/53
DESCRIPTION:Title: Reversible Markov chains with nonnegative spectrum\nby Ro
berto Oliveira (IMPA) as part of Oxford discrete mathematics and probabili
ty seminar\n\n\nAbstract\nThe title of the talk corresponds to a family of
interesting random processes\, which includes lazy random walks on graphs
and much beyond them. Often\, a key step in analysing such processes is t
o estimate their spectral gaps (ie. the difference between two largest eig
envalues). It is thus of interest to understand what else about the chain
we can know from the spectral gap. We will present a simple comparison ide
a that often gives us the best possible estimates. In particular\, we re-o
btain and improve upon several known results on hitting\, meeting\, and in
tersection times\; return probabilities\; and concentration inequalities f
or time averages. We then specialize to the graph setting\, and obtain sha
rp inequalities in that setting. This talk is based on work that has been
in progress for far too long with Yuval Peres.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Annika Heckel (LMU München)
DTSTART:20210504T130000Z
DTEND:20210504T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/54
DESCRIPTION:Title: How does the chromatic number of a random graph vary?\nby
Annika Heckel (LMU München) as part of Oxford discrete mathematics and p
robability seminar\n\n\nAbstract\nHow much does the chromatic number of th
e random graph $G(n\, 1/2)$ vary? Shamir and Spencer proved that it is con
tained in some sequence of intervals of length about $n^{1/2}$. Alon impro
ved this slightly to $n^{1/2} / \\log n$. Until recently\, however\, no lo
wer bounds on the fluctuations of the chromatic number of $G(n\, 1/2)$ wer
e known\, even though the question was raised by Bollobás many years ago.
I will talk about the main ideas needed to prove that\, at least for infi
nitely many $n$\, the chromatic number of $G(n\, 1/2)$ is not concentrated
on fewer than $n^{1/2-o(1)}$ consecutive values.\nI will also discuss the
Zigzag Conjecture\, made recently by Bollobás\, Heckel\, Morris\, Panagi
otou\, Riordan and Smith: this proposes that the correct concentration int
erval length 'zigzags' between $n^{1/4+o(1)}$ and $n^{1/2+o(1)}$\, depend
ing on $n$.\nJoint work with Oliver Riordan.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jean-François Le Gall (Paris-Saclay)
DTSTART:20210504T143000Z
DTEND:20210504T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/55
DESCRIPTION:Title: Geodesics in random geometry\nby Jean-François Le Gall (
Paris-Saclay) as part of Oxford discrete mathematics and probability semin
ar\n\n\nAbstract\nWe discuss the behavior of geodesics in the continuous m
odels of random geometry known as the Brownian map and the Brownian plane.
We say that a point $x$ is a geodesic star with $m$ arms if $x$ is the en
dpoint of $m$ disjoint geodesics. We prove that the set of all geodesic st
ars with $m$ arms has dimension $5-m$\, for $m=1\,2\,3\,4$. This complemen
ts recents results of Miller and Qian\, who derived upper bounds for these
dimensions.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cécile Mailler (Bath)
DTSTART:20210511T140000Z
DTEND:20210511T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/56
DESCRIPTION:Title: The ants walk: finding geodesics in graphs using reinforcemen
t learning\nby Cécile Mailler (Bath) as part of Oxford discrete mathe
matics and probability seminar\n\n\nAbstract\nHow does a colony of ants fi
nd the shortest path between its nest and a source of food without any mea
ns of communication other than the pheromones each ant leave behind itself
?\nIn this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseil
le)\, we introduce a new probabilistic model for this phenomenon. In this
model\, the nest and the source of food are two marked nodes in a finite g
raph. Ants perform successive random walks from the nest to the food\, and
the distribution of the $n$th walk depends on the trajectories of the $(n
-1)$ previous walks through some linear reinforcement mechanism.\nUsing st
ochastic approximation methods\, couplings with Pólya urns\, and the elec
tric conductances method for random walks on graphs (which I will explain
on some simple examples)\, we prove that\, depending on the exact reinforc
ement rule\, the ants may or may not always find the shortest path(s) betw
een their nest and the source food.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Asaf Ferber (University of California\, Irvine)
DTSTART:20210511T153000Z
DTEND:20210511T163000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/57
DESCRIPTION:Title: Lower bounds for multicolor Ramsey numbers\nby Asaf Ferbe
r (University of California\, Irvine) as part of Oxford discrete mathemati
cs and probability seminar\n\n\nAbstract\nWe present an exponential improv
ement to the lower bound on diagonal Ramsey numbers for any fixed number o
f colors greater than two.\nThis is a joint work with David Conlon.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mihyun Kang (Graz)
DTSTART:20210518T130000Z
DTEND:20210518T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/58
DESCRIPTION:Title: Benjamini-Schramm local limits of sparse random planar graphs
\nby Mihyun Kang (Graz) as part of Oxford discrete mathematics and pro
bability seminar\n\n\nAbstract\nIn this talk we will discuss some classica
l and recent results on local limits of random graphs. It is well known th
at the limiting object of the local structure of the classical Erdos-Renyi
random graph is a Galton-Watson tree. This can nicely be formalised in th
e language of Benjamini-Schramm or Aldous-Steele local weak convergence. R
egarding local limits of sparse random planar graphs\, there is a smooth t
ransition from a Galton-Watson tree to a Skeleton tree. This talk is based
on joint work with Michael Missethan.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vincent Tassion (ETH)
DTSTART:20210525T130000Z
DTEND:20210525T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/59
DESCRIPTION:Title: Crossing probabilities for planar percolation\nby Vincent
Tassion (ETH) as part of Oxford discrete mathematics and probability semi
nar\n\n\nAbstract\nPercolation models were originally introduced to descri
be the propagation of a fluid in a random medium. In dimension two\, the p
ercolation properties of a model are encoded by so-called crossing probabi
lities (probabilities that certain rectangles are crossed from left to rig
ht). In the eighties\, Russo\, Seymour and Welsh obtained general bounds o
n crossing probabilities for Bernoulli percolation (the most studied perco
lation model\, where edges of a lattice are independently erased with some
given probability $1-p$). These inequalities rapidly became central tools
to analyze the critical behavior of the model.\nIn this talk I will prese
nt a new result which extends the Russo-Seymour-Welsh theory to general pe
rcolation measures satisfying two properties: symmetry and positive correl
ation. This is a joint work with Laurin Köhler-Schindler.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Krivelevich (Tel Aviv)
DTSTART:20210525T143000Z
DTEND:20210525T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/60
DESCRIPTION:Title: Cycle lengths in sparse random graphs\nby Michael Krivele
vich (Tel Aviv) as part of Oxford discrete mathematics and probability sem
inar\n\n\nAbstract\nWe study the set $L(G)$ of cycle lengths that appear i
n a sparse binomial random graph $G(n\,c/n)$ and in a random $d$-regular g
raph $G_{n\,d}$. We show in particular that for most values of $c$\, for $
G$ drawn from $G(n\,c/n)$ the set $L(G)$ contains typically an interval $[
\\omega(1)\, (1-o(1))L_{\\max}(G)]$\, where $L_{\\max}(G)$ is the length o
f a longest cycle (the circumference) of $G$. For the case of random $d$-r
egular graphs\, $d\\geq 3$ fixed\, we obtain an accurate asymptotic estim
ate for the probability of $L(G)$ to contain a full interval $[k\,n]$ for
a fixed $k\\geq 3$. Similar results are obtained also for the supercritica
l case $G(n\,(1+\\epsilon)/n)$\, and for random directed graphs.\nA joint
work with Yahav Alon and Eyal Lubetzky.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Konstantin Tikhomirov (Georgia Tech)
DTSTART:20210601T133000Z
DTEND:20210601T143000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/61
DESCRIPTION:Title: Invertibility of random square matrices\nby Konstantin Ti
khomirov (Georgia Tech) as part of Oxford discrete mathematics and probabi
lity seminar\n\n\nAbstract\nConsider an $n$ by $n$ random matrix $A$ with
i.i.d entries. In this talk\, we discuss some results on the magnitude of
the smallest singular value of $A$\, and\, in particular\, the problem of
estimating the singularity probability when the entries of $A$ are discret
e.\n\nJoint with Oxford's Random Matrix Theory Seminar.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gérard Ben Arous (Courant Institute)
DTSTART:20210601T143000Z
DTEND:20210601T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/62
DESCRIPTION:Title: Random Determinants and the Elastic Manifold\nby Gérard
Ben Arous (Courant Institute) as part of Oxford discrete mathematics and p
robability seminar\n\n\nAbstract\nThis is joint work with Paul Bourgade an
d Benjamin McKenna (Courant Institute\, NYU).\nThe elastic manifold is a p
aradigmatic representative of the class of disordered elastic systems. The
se models describe random surfaces with rugged shapes resulting from a com
petition between random spatial impurities (preferring disordered configur
ations)\, on the one hand\, and elastic self-interactions (preferring orde
red configurations)\, on the other. The elastic manifold model is interest
ing because it displays a depinning phase transition and has a long histor
y as a testing ground for new approaches in statistical physics of disorde
red media\, for example for fixed dimension by Fisher (1986) using functio
nal renormalization group methods\, and in the high-dimensional limit by M
ézard and Parisi (1992) using the replica method.\nWe study the topology
of the energy landscape of this model in the Mézard-Parisi setting\, and
compute the (annealed) topological complexity both of total critical point
s and of local minima. Our main result confirms the recent formulas by Fyo
dorov and Le Doussal (2020) and allows to identify the boundary between si
mple and glassy phases. The core argument relies on the analysis of the as
ymptotic behavior of large random determinants in the exponential scale.\n
\nJoint with Oxford's Random Matrix Theory Seminar.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Amedeo Sgueglia (LSE)
DTSTART:20210518T143000Z
DTEND:20210518T153000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/63
DESCRIPTION:Title: Factors in randomly perturbed graphs\nby Amedeo Sgueglia
(LSE) as part of Oxford discrete mathematics and probability seminar\n\n\n
Abstract\nWe study the model of randomly perturbed dense graphs\, which is
the union of any $n$-vertex graph $G_\\alpha$ with minimum degree at leas
t $\\alpha n$ and the binomial random graph $G(n\,p)$. In this talk\, we s
hall examine the following central question in this area: to determine whe
n $G_\\alpha \\cup G(n\,p)$ contains $H$-factors\, i.e. spanning subgrap
hs consisting of vertex disjoint copies of the graph $H$. We offer several
new sharp and stability results.\nThis is joint work with Julia Böttcher
\, Olaf Parczyk\, and Jozef Skokan.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sumit Mukherjee (Columbia)
DTSTART:20211012T130000Z
DTEND:20211012T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/64
DESCRIPTION:Title: Generalized birthday problem for October 12\nby Sumit Muk
herjee (Columbia) as part of Oxford discrete mathematics and probability s
eminar\n\n\nAbstract\nSuppose there are $n$ students in a class. But assum
e that not everybody is friends with everyone else\, and there is a graph
which determines the friendship structure. What is the chance that there a
re two friends in this class\, both with birthdays on October 12? More gen
erally\, given a simple labelled graph $G_n$ on $n$ vertices\, color each
vertex with one of $c=c_n$ colors chosen uniformly at random\, independent
from other vertices. We study the question: what is the number of monochr
omatic edges of color 1?\n\nAs it turns out\, the limiting distribution ha
s three parts\, the first and second of which are quadratic and linear fun
ctions of a homogeneous Poisson point process\, and the third component is
an independent Poisson. In fact\, we show that any distribution limit mus
t belong to the closure of this class of random variables. As an applicati
on\, we characterize exactly when the limiting distribution is a Poisson r
andom variable.\n\nThis talk is based on joint work with Bhaswar Bhattacha
rya and Somabha Mukherjee.\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matija Bucić (Princeton/IAS)
DTSTART:20211109T140000Z
DTEND:20211109T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/65
DESCRIPTION:by Matija Bucić (Princeton/IAS) as part of Oxford discrete ma
thematics and probability seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mariana Olvera-Cravioto (UNC Chapel Hill)
DTSTART:20211123T140000Z
DTEND:20211123T150000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/66
DESCRIPTION:by Mariana Olvera-Cravioto (UNC Chapel Hill) as part of Oxford
discrete mathematics and probability seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashwin Sah (MIT)
DTSTART:20211026T130000Z
DTEND:20211026T140000Z
DTSTAMP:20260404T095501Z
UID:OxfordDMProb/67
DESCRIPTION:by Ashwin Sah (MIT) as part of Oxford discrete mathematics and
probability seminar\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/OxfordDMProb/67/
END:VEVENT
END:VCALENDAR