\n

\n As a concrete example\, consider a sample A of Ising model on a rooted tree (re gular\, Galton-Watson\, etc). Let B be a noisy version of A obtained by in dependently perturbing each spin as follows: $B_v$ equals to $A_v$ with so me small probability $\\delta$ and otherwise taken to be a uniform +-1 (al ternatively\, 0). We show that the distribution of the root spin $A_\\rh o$ conditioned on values $B_v$ of all vertices $v$ at a large distance fro m the root is independent of $\\delta$ and coincides with $\\delta=0$. Pr eviously this was only known for sufficiently ``low-temperature'' models. Our proof consists of constructing a metric under which the BP operator is a contraction (albeit non-multiplicative). I hope to convince you our pro of is technically rather simple.\n

\n

\n Th is simultaneously closes the following 5 conjectures in the literature:\n

\n

\n

\n Joint work with Qian Yu (Princeton).\n LOCATION:https://researchseminars.org/talk/Probability/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Marcelo Hilario (UFMG) DTSTART;VALUE=DATE-TIME:20211101T201500Z DTEND;VALUE=DATE-TIME:20211101T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/40 DESCRIPTION:Title: Random walks on dynamic random environments with non-uniform mixing.< /a>\nby Marcelo Hilario (UFMG) as part of MIT probability seminar\n\n\nAbs tract\nIn this talk\, we will discuss recent results on the limiting behav ior of random walks in dynamic random environments. We will mainly discuss the case when the random walk evolves on one-dimensional random environme nts given by conservative interacting particle systems such as the simple symmetric exclusion process. Its transitions probabilities will depend on the current occupation environment nearby. Conservation of particles leads to poor mixing conditions and we explain how renormalization techniques c an be useful to obtain the law of large numbers\, large deviation estimate s\, and sometimes central limit theorems. The talk is based on several joi nt works with Oriane Blondel\, Frank den Hollander\, Daniel Kious\, Renato dos Santos\, Vladas Sidoravicius and Augusto Teixeira.\n LOCATION:https://researchseminars.org/talk/Probability/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Nishant Changotia (Tata Institute of Fundamental Research) DTSTART;VALUE=DATE-TIME:20211213T211500Z DTEND;VALUE=DATE-TIME:20211213T221500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/41 DESCRIPTION:Title: Title to be announced\nby Nishant Changotia (Tata Institute of Fu ndamental Research) as part of MIT probability seminar\n\n\nAbstract\nAbst ract to be shared\n LOCATION:https://researchseminars.org/talk/Probability/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Cole Graham (Brown University) DTSTART;VALUE=DATE-TIME:20220214T211500Z DTEND;VALUE=DATE-TIME:20220214T221500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/42 DESCRIPTION:Title: Stationary measures for stochastic conservation laws\nby Cole Gra ham (Brown University) as part of MIT probability seminar\n\n\nAbstract\n\ \noindent At long times\, many SPDEs relax to statistically steady states. In this talk\, I will consider the existence and uniqueness of such stati onary measures for stochastically-forced viscous conservation laws on the line. A special case\, the stochastic Burgers equation\, has received a gr eat deal of attention due to its links to the KPZ and stochastic heat equa tions. Stochastic Burgers is known to admit a unique spacetime-stationary ergodic measure for each mean. However\, existing proofs rely on the Cole –Hopf transformation\, which does not extend to other conservation laws. I will discuss a comparison-based approach that recovers partial results for more general conservation laws. In particular\, such SPDEs admit infin itely many stationary ergodic measures\, and there is at most one such mea sure for each mean. \\\\\n\\vspace{2ex}\n\\noindent This is joint work wit h Theodore Drivas\, Alexander Dunlap\, Joonhyun La\, and Lenya Ryzhik.\n LOCATION:https://researchseminars.org/talk/Probability/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Mark Sellke (Stanford University) DTSTART;VALUE=DATE-TIME:20220307T211500Z DTEND;VALUE=DATE-TIME:20220307T221500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/43 DESCRIPTION:Title: Algorithmic Thresholds in Mean-Field Spin Glasses\nby Mark Sellke (Stanford University) as part of MIT probability seminar\n\nLecture held in Room 2-147 in the Simons Building.\n\nAbstract\n\\noindent I will expla in recent progress on computing approximate ground states of mean-field sp in glass Hamiltonians\, which are certain random functions in high dimensi on. While the asymptotic ground state energy OPT is given by the famous Pa risi formula\, the landscape of these functions often include many bad loc al optima which impede optimization by efficient algorithms. In the positi ve direction\, I will explain algorithms based on approximate message pass ing which asymptotically achieve a value ALG given by an extended Parisi f ormula. The case ALG=OPT has a "no overlap gap" or "full replica symmetry breaking" interpretation\, but in general these algorithms may fail to rea ch asymptotic optimality. In the negative direction\, I will explain why n o algorithm with suitably Lipschitz dependence on the random disorder can surpass the threshold ALG. This result applies to many standard optimizati on algorithms\, such as gradient descent and its variants on dimension-fre e time scales. Based on joint works with Ahmed El Alaoui\, Brice Huang\, a nd Andrea Montanari.\n LOCATION:https://researchseminars.org/talk/Probability/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Oanh Nguyen (Brown University) DTSTART;VALUE=DATE-TIME:20220328T201500Z DTEND;VALUE=DATE-TIME:20220328T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/44 DESCRIPTION:Title: Survival time of the contact process on random graphs\nby Oanh Ng uyen (Brown University) as part of MIT probability seminar\n\nLecture held in Room 2-147 in the Simons Building.\n\nAbstract\n\\noindent The contact process is a model for the spread of infections on graphs. In this talk\, we discuss the contact process on random graphs with low infection rate $ \\lambda$. For random $d$-regular graphs\, it is known that the survival t ime is $O(\\log n)$ below the critical $\\lambda_c$. By contrast\, on the Erdos-Renyi random graphs $G(n\,d/n)$\, rare high degree vertices result in much longer survival times. We show that the survival time is governed by high density local configurations\, in particular large connected compo nents of high degree vertices on which the infection lasts for time $n^{\\ lambda^{2+o(1)}}$. We shall discuss how to obtain a matching upper bound. Our methods\, moreover\, generalize to random graphs with given degree di stributions that have exponential moments.\\\\\n\\vspace{2ex}\n\\noindent Joint work with Allan Sly. \\\\\n LOCATION:https://researchseminars.org/talk/Probability/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Johannes Alt (Courant Institute) DTSTART;VALUE=DATE-TIME:20220404T201500Z DTEND;VALUE=DATE-TIME:20220404T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/45 DESCRIPTION:Title: Localization and Delocalization in Erdős–Rényi graphs\nby Joh annes Alt (Courant Institute) as part of MIT probability seminar\n\nLectur e held in Room 2-147 in the Simons Building.\n\nAbstract\nWe consider the Erdős–Rényi graph on N vertices with edge probability d/N. It is well known that the structure of this graph changes drastically when d is of or der log N. Below this threshold it develops inhomogeneities which lead to the emergence of localized eigenvectors\, while the majority of the eigenv ectors remains delocalized. In this talk\, I will present the phase diagra m depicting these localized and delocalized phases and our recent progress in establishing it rigorously.\n\nThis is based on joint works with Rapha el Ducatez and Antti Knowles.\n LOCATION:https://researchseminars.org/talk/Probability/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Sky Cao (Stanford University) DTSTART;VALUE=DATE-TIME:20220411T201500Z DTEND;VALUE=DATE-TIME:20220411T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/46 DESCRIPTION:Title: Exponential decay of correlations in finite gauge group lattice gauge theories\nby Sky Cao (Stanford University) as part of MIT probability seminar\n\nLecture held in Room 2-147 in the Simons Building.\n\nAbstract \nLattice gauge theories with finite gauge groups are statistical mechanic al models\, very much akin to the Ising model\, but with some twists. In t his talk\, I will describe how to show exponential decay of correlations f or these models at low temperatures. This is based on joint work with Arka Adhikari.\n LOCATION:https://researchseminars.org/talk/Probability/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrew Ahn (Cornell University) DTSTART;VALUE=DATE-TIME:20220425T201500Z DTEND;VALUE=DATE-TIME:20220425T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/47 DESCRIPTION:Title: Lyapunov Exponents of Random Matrix Products and Brownian Motion on G L(n\,C)\nby Andrew Ahn (Cornell University) as part of MIT probability seminar\n\nLecture held in Room 2-147 in the Simons Building.\n\nAbstract \nConsider the discrete-time process formed by the singular values of prod ucts of random matrices\, where time corresponds to the number of matrix f actors. It is known due to Oseledets' theorem that under general assumptio ns\, the Lyapunov exponents converge as the number of matrix factors tend to infinity. In this talk\, we consider random matrices with distributiona l invariance under right multiplication by unitary matrices\, which includ e Ginibre matrices and truncated unitary matrices. The corresponding singu lar value process is Markovian with additional structure that admits study via integrable probability techniques. In this talk\, I will discuss rece nt results on the Lyapunov exponents in the setting where the number M mat rix factors tend to infinity simultaneously with matrix sizes N. When this limit is tuned so that M and N grow on the same order\, the limiting Lyap unov exponents can be described in terms of Dyson Brownian motion with a s pecial drift vector\, which in turn can be linked to a matrix-valued diffu sion on the complex general linear group. We find that this description is universal\, under general assumptions on the spectrum of the matrix facto rs.\n LOCATION:https://researchseminars.org/talk/Probability/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Massimiliano Gubinelli (University of Bonn) DTSTART;VALUE=DATE-TIME:20220502T201500Z DTEND;VALUE=DATE-TIME:20220502T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/48 DESCRIPTION:Title: What is stochastic quantization?\nby Massimiliano Gubinelli (Univ ersity of Bonn) as part of MIT probability seminar\n\n\nAbstract\nIn this talk I will introduce the idea of stochastic\nquantization from a mathemat ical perspective\, that is as a tool to\nanalyze rigorously Euclidean quan tum fields. I will show that there\nare several different "stochastic quan tizations” for which we will\nidentify common structures and ideas which take the form of a\nstochastic analysis of Euclidean quantum fields.\n LOCATION:https://researchseminars.org/talk/Probability/48/ END:VEVENT BEGIN:VEVENT SUMMARY:Phil Sosoe (Cornell University) DTSTART;VALUE=DATE-TIME:20220509T190000Z DTEND;VALUE=DATE-TIME:20220509T200000Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/49 DESCRIPTION:Title: Almost-optimal regularity conditions in the CLT for Wigner matrices.< /a>\nby Phil Sosoe (Cornell University) as part of MIT probability seminar \n\nLecture held in Room 2-361 in the Simons Building.\n\nAbstract\nWe con sider linear spectral statistics for test functions of low regularity and Wigner matrices with smooth entry distribution. We show that for functions in the Sobolev space $H^{1/2 + \\epsilon}$ or the space $C^{1/2 + \\epsil on}$ that are supported within the spectral bulk of the semicircle distrib ution\, the variance remains bounded asymptotically. As a consequence\, th ese linear spectral statistics have asymptotic Gaussian fluctuations with the same variance as in the CLT for functions of higher regularity\, for a ny $\\epsilon > 0$. This result is nearly optimal in the sense that the va riance does remain bounded for functions in $H^{1/2}$\, and was previously known only for matrices in Gaussian Unitary Ensemble.\n LOCATION:https://researchseminars.org/talk/Probability/49/ END:VEVENT BEGIN:VEVENT SUMMARY:Hugo Falconet (NYU) DTSTART;VALUE=DATE-TIME:20220228T211500Z DTEND;VALUE=DATE-TIME:20220228T221500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/50 DESCRIPTION:Title: Metric growth dynamics in Liouville quantum gravity\nby Hugo Falc onet (NYU) as part of MIT probability seminar\n\n\nAbstract\n\\noindent Li ouville quantum gravity (LQG) is a canonical model of random geometry. Ass ociated with the planar Gaussian free field\, this geometry with special c onformal symmetries was introduced in the physics literature by Polyakov i n the 80's and is conjectured to describe the scaling limit of random plan ar maps. In this talk\, I will introduce LQG as a metric measure space and discuss recent results on the associated metric growth dynamics. The prim ary focus will be on the dynamics of the trace of the free field on the bo undary of growing LQG balls. \\\\\n\\vspace{2ex}\n\\noindent Based on a jo int work with Julien Dubédat.\n LOCATION:https://researchseminars.org/talk/Probability/50/ END:VEVENT BEGIN:VEVENT SUMMARY:Eviatar Procaccia (Technion) DTSTART;VALUE=DATE-TIME:20220314T170000Z DTEND;VALUE=DATE-TIME:20220314T180000Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/51 DESCRIPTION:Title: Stationary Hastings-Levitov model\nby Eviatar Procaccia (Technion ) as part of MIT probability seminar\n\nLecture held in Room 2-132 in the Simons Building.\n\nAbstract\nWe construct and study a stationary version of the Hastings-Levitov(0) model. We prove that unlike the classical model \, in the stationary case\, particle sizes are tight\, yielding that this model can be seen as a tractable off-lattice Diffusion Limited Aggregation (DLA). The stationary setting together with a geometric interpretations o f the harmonic measure yields new geometric results such as finiteness of arms\, exact growth rate and fractal dimension equals 3/2\, corresponding to a numerical prediction of Meakin from 1983 for the gyration radius of D LA growing on a long line segment. We will also show that similar results can be achieved in a cylinder.\n LOCATION:https://researchseminars.org/talk/Probability/51/ END:VEVENT BEGIN:VEVENT SUMMARY:Dominik Schmid (Princeton University) DTSTART;VALUE=DATE-TIME:20220314T201500Z DTEND;VALUE=DATE-TIME:20220314T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/52 DESCRIPTION:Title: Mixing times for the TASEP on the circle\nby Dominik Schmid (Prin ceton University) as part of MIT probability seminar\n\nLecture held in Ro om 2-147 in the Simons Building.\n\nAbstract\nThe exclusion process is one of the best-studied examples of an interacting particle system. In this t alk\, we consider simple exclusion processes on finite graphs. We give an overview over some recent results on the mixing time of the totally asymme tric simple exclusion process (TASEP). In particular\, we provide bounds o n the mixing time of the TASEP on the circle\, using a connection to perio dic last passage percolation. This talk is based on joint work with Allan Sly (Princeton).\n LOCATION:https://researchseminars.org/talk/Probability/52/ END:VEVENT BEGIN:VEVENT SUMMARY:Remi Rhodes (Aix-Marseille Université) DTSTART;VALUE=DATE-TIME:20220425T170000Z DTEND;VALUE=DATE-TIME:20220425T180000Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/53 DESCRIPTION:Title: Segal’s axioms and conformal bootstrap in Liouville theory\nb y Remi Rhodes (Aix-Marseille Université) as part of MIT probability semin ar\n\nLecture held in Room 2 - 361 in the Simons Building.\n\nAbstract\nCo nformal field theories (CFT) are expected to describe models of statistica l physics in 2D undergoing a second order phase transition at their critic al point. Several axiomatics have been proposed to lay the mathematical fo undations for the concept of CFT. In Segal’s approach\, the data for a CFT are an Hilbert space H and a map that associates to each Riemann surfa ce S with boundary a Hilbert-Schmidt operator (called amplitude) acting on the tensor product $H^b$ with b the number of boundary components of S. A mplitudes are then assumed to compose in a natural way under gluing of sur faces along their boundaries. Segal’s approach turned out to be very att ractive for mathematicians in view of its geometric flavor. Also\, it give s a geometrical way to understand the conformal bootstrap conjecture in ph ysics: correlation functions of CFT should factorize as an integral over t heir spectrum of the product of (squared) conformal blocks times the struc ture constants of the CFT (the 3 point correlation functions on the Rieman n sphere). Conformal blocks are holomorphic functions of the moduli of the space of Riemann surfaces with marked point\, which are universal in the sense that they only depend on the commutation relations of a given Lie al gebra\, the Virasoro algebra. Structure constants are model dependent. In this talk I will explain how this picture for CFTs drawn by Segal applies to Liouville theory (LCFT)\, which is a non rational conformal field theo ry developed in the early 80s in physics to describe random Riemannian metrics on Riemann surfaces.\n LOCATION:https://researchseminars.org/talk/Probability/53/ END:VEVENT BEGIN:VEVENT SUMMARY:Guilherme Silva (University of Sao Paolo) DTSTART;VALUE=DATE-TIME:20220912T201500Z DTEND;VALUE=DATE-TIME:20220912T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/54 DESCRIPTION:Title: Universality for a class of statistics of Hermitian random matrices a nd the integro-differential Painlevé II equation.\nby Guilherme Silva (University of Sao Paolo) as part of MIT probability seminar\n\n\nAbstrac t\nIt has been known since the 1990s that fluctuations of eigenvalues of r andom matrices\, when appropriately scaled and in the sense of one-point d istribution\, converge to the Airy2 point process in the large matrix limi t. In turn\, the latter can be described by the celebrated Tracy-Widom dis tribution.\n\nIn this talk we discuss recent findings of Ghosal and myself \, showing that certain statistics of eigenvalues also converge universali ty to appropriate statistics of the Airy2 point process\, interpolating be tween a hard and soft edge of eigenvalues. Such found statistics connect a lso to the integro-differential Painlevé II equation\, in analogy with th e celebrated Tracy-Widom connection between Painlevé II and the Airy2 pro cess.\n LOCATION:https://researchseminars.org/talk/Probability/54/ END:VEVENT BEGIN:VEVENT SUMMARY:Matteo Mucciconi (University of Warwick) DTSTART;VALUE=DATE-TIME:20220919T201500Z DTEND;VALUE=DATE-TIME:20220919T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/55 DESCRIPTION:Title: Title to be announced\nby Matteo Mucciconi (University of Warwick ) as part of MIT probability seminar\n\nAbstract: TBA\n LOCATION:https://researchseminars.org/talk/Probability/55/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Hough (Stony Brook University) DTSTART;VALUE=DATE-TIME:20221031T201500Z DTEND;VALUE=DATE-TIME:20221031T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/56 DESCRIPTION:Title: Covering systems of congruences\nby Robert Hough (Stony Brook Uni versity) as part of MIT probability seminar\n\n\nAbstract\n\\noindent A di stinct covering system of congruences is a list of congruences\n\\[\na_i \ \bmod m_i\, \\qquad i = 1\, 2\, ...\, k\n\\]\nwhose union is the integers. Erd\\H{o}s asked if the least modulus $m_1$ of a distinct covering syste m of congruences can be arbitrarily large (the minimum modulus problem for covering systems\, \\$ 1000 ) and if there exist distinct covering system s of congruences all of whose moduli are odd (the odd problem for covering systems\, \\$ 25). I'll discuss my proof of a negative answer to the min imum modulus problem\, and a quantitative refinement with Pace Nielsen tha t proves that any distinct covering system of congruences has a modulus di visible by either 2 or 3. The proofs use the probabilistic method and in particular use a sequence of pseudorandom probability measures adapted to the covering process. Time permitting\, I may briefly discuss a reformula tion of our method due to Balister\, Bollob\\'{a}s\, Morris\, Sahasrabudhe and Tiba which solves a conjecture of Shinzel (any distinct covering syst em of congruences has one modulus that divides another) and gives a negati ve answer to the square-free version of the odd problem.\n LOCATION:https://researchseminars.org/talk/Probability/56/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexandros Eskenazis (CNRS) DTSTART;VALUE=DATE-TIME:20221003T201500Z DTEND;VALUE=DATE-TIME:20221003T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/57 DESCRIPTION:Title: Learning low-degree functions on the discrete hypercube\nby Alexa ndros Eskenazis (CNRS) as part of MIT probability seminar\n\nLecture held in Room 2-147 in the Simons Building.\n\nAbstract\nLet f be an unknown fun ction on the n-dimensional discrete hypercube. How many values of f do we need in order to approximately reconstruct the function? In this talk we s hall discuss the random query model for this fundamental problem from comp utational learning theory. We will explain a newly discovered connection w ith a family of polynomial inequalities going back to Littlewood (1930) wh ich will in turn allow us to derive sharper estimates for the the query co mplexity of this model\, exponentially improving those which follow from t he classical Low-Degree Algorithm of Linial\, Mansour and Nisan (1989). Ti me permitting\, we will also show a matching information-theoretic lower b ound. Based on joint works with Paata Ivanisvili (UC Irvine) and Lauritz S treck (Cambridge).\n LOCATION:https://researchseminars.org/talk/Probability/57/ END:VEVENT BEGIN:VEVENT SUMMARY:Hao Shen (UW-Madison) DTSTART;VALUE=DATE-TIME:20221024T201500Z DTEND;VALUE=DATE-TIME:20221024T211500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/58 DESCRIPTION:Title: Stochastic Yang-Mills process in 2D and 3D.\nby Hao Shen (UW-Madi son) as part of MIT probability seminar\n\nLecture held in Room 2-147 in t he Simons Building.\n\nAbstract\nWe will discuss stochastic quantization o f the Yang-Mills model on two and three dimensional torus. In stochastic q uantization we consider the Langevin dynamic for the Yang-Mills model whic h is described by a stochastic PDE. We construct local solution to this SP DE and prove that the solution has a gauge invariant property in law\, whi ch then defines a Markov process on the space of gauge orbits. We will als o describe the construction of this orbit space\, on which we have well-de fined holonomies and Wilson loop observables. Based on joint work with Aja y Chandra\, Ilya Chevyrev\, and Martin Hairer.\n LOCATION:https://researchseminars.org/talk/Probability/58/ END:VEVENT BEGIN:VEVENT SUMMARY:Dor Elboim (Princeton University) DTSTART;VALUE=DATE-TIME:20221107T211500Z DTEND;VALUE=DATE-TIME:20221107T221500Z DTSTAMP;VALUE=DATE-TIME:20230610T051021Z UID:Probability/59 DESCRIPTION:Title: Infinite cycles in the interchange process in five dimensions\nby Dor Elboim (Princeton University) as part of MIT probability seminar\n\n\ nAbstract\n\\noindent In the interchange process on a graph G=(V\,E)\, the re is a particle on each vertex of the graph and an independent Poisson cl ock on each one of the edges. Once the clock of an edge rings\, the two pa rticles on the two sides of the edge switch. After time t\, the particles are permuted according to a random permutation $\\pi_t:V\\to V$. A famous conjecture of Balint Toth states that the following holds when $G=\\mathbb $ $Z^d$ :\n(1) If d=2\, then the permutation $\\pi_t$ contains only finite cycles for all t>0.\n(2) If $d\\ge 3$\, then there exists $t_c$ such that for $t